High School
SAT
ACT
University
MCAT
LSAT
DAT

Projection onto a subspace

Related Topics

Wize University Linear Algebra Textbook > Orthogonality

Orthogonal Complement and Orthogonal Projection (COMING SOON)

3 Activities

Wize University Linear Algebra Textbook > Linear Transformations

Linear Transformations

5 Activities

checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
Suppose u⃗,v⃗∈Rn\vec u,\vec v\in\mathbb{R}^n satisfy u⃗⋅v⃗=0,∣∣u⃗∣∣=∣∣v⃗∣∣=1\vec u\cdot\vec v=0, ||\vec u||=||\vec v||=1, and define U:=span{u⃗,v⃗}U := span\{\vec u,\vec v\} .

Using the definition that projUx⃗=A(ATA)−1ATx⃗proj_U\vec x=A(A^TA)^{-1}A^T\vec x , where A=[u⃗v⃗]A=\left[\begin{array}{rr}\vec u&\vec v\end{array}\right] , show that for all x⃗∈Rn\vec x\in\mathbb{R}^n we have that projUx⃗=proju⃗x⃗+projv⃗x⃗proj_U\vec x=proj_{\vec u}\vec x+proj_{\vec v}\vec x .
More Orthogonal Complement and Orthogonal Projection (COMING SOON) Questions:
19.4F_Final_Builder_Ch_17.3_Least_Squares_$\tkco{eg4}$_$\key{Final}$_Builder_$\tkcth{17.3.}\tkcf{4}$_
The figure below illustrates the relationship between the Im (A‾)\bct{\text{Im}\, (\A)} and b⃗\bco{\color{cyan}\vb} for the system given by:
{x−2y=−42x−4y=7⇔[  1−2    2−4  ][abc] ⁣ ⁣ ⁣ ⁣⏟:= A‾[xy]=[−47][abc]⏟:= b⃗ \left\{ \begin{array}{rr} x - 2y \quad =& -4 \\ 2x - 4y \quad =& 7 \end{array} \right. % \qquad \Leftrightarrow \qquad % \underbrace{ \begin{bmatrix} \; 1 & -2 \; \\ \; 2 & -4 \; \end{bmatrix} \vphantom{\colvecth{a}{b}{c}\!\!\!\!} }_{:= \, \bct{\A}} % \colvec{x}{y} % = % \underbrace{ \colvec{-4}{7} \vphantom{\colvecth{a}{b}{c}} }_{:=\, \bco{\color{cyan}\vb}}
Let b⃗Im(A‾)=proj⇀Im(A‾)(b⃗)\vb_{_{\textrm{Im}(\A)}} = \overrightharpoon{\text{proj}}_{_{\textrm{Im}(\A)}}(\vb), i.e. the (vector) component of b⃗\vb which is in the image of A‾\A. Identify this vector on the figure; we'll call this component b⃗∣∣\vb_{_{||}} for short.
The projection of a vector b⃗\vb onto the image of a set of vectors {v⃗1,v⃗2}\big\{ \vv_1, \vv_2 \big\} is equal to the sum of the projection of the vector b⃗\vb onto each of the individual vectors, i.e.:
proj⇀Im(v⃗1,v⃗2)(b⃗)=proȷ⇀v1⃗ ⁣(b⃗)+proȷ⇀v2⃗ ⁣(b⃗) \overrightharpoon{\textrm{proj}}_{_{\textrm{Im}(\vv_1,\vv_2)}}(\vb) = \proj{b}{v_1} + \proj{b}{v_2}
Span
Let B:={(1,1,0,2),(0,1,0,−1),(0,0,1,1)}B:=\{(1,1,0,2),(0,1,0,-1),(0,0,1,1)\} , and U:=spanBU:=spanB .
a) Find a pairwise orthogonal set of nonzero vectors OO so that U=spanOU=spanO .
b) Write the Fourier expansion of (0,0,1,0)(0,0,1,0) in terms of the basis OO .
19.4F_Final_Builder_Ch_17.2_Orthogonal_Complement_$\tkco{eg4}$_$\key{Final}$_Builder_$\tkcth{17.2.}\tkcf{4}$_
Suppose U={v⃗1, … , v⃗n}U = \left\{ \vv_1,\, \dots\,,\, \vv_n \right\} is an orthogonal basis for Rn\mathbb{R}^{n}, then any vector w⃗\vw in Rn\mathbb{R}^{n} can be written as a linear combination of these vectors, i.e.
w⃗=∑inci v⃗i=c1 v⃗1+c2 v⃗2+ … +cn v⃗n. \vw = \sum_{i}^{n} c_i \, \vv_i = c_1 \, \vv_1 + c_2 \, \vv_2 + \, \dots \, + c_n \, \vv_n.
Find an expression for the constants c1c_1, c2c_2, …\dots, cnc_n in terms of w⃗\vw and the vectors in UU.
19.4F_WML_6_$\tkco{eg10}$_$\key{Final}$_Builder_$\tkcth{17.1.}\tkct{11}$_
Find the set of all vectors perpendicular to the vectors {a⃗1, a⃗2 }={[1101], [0111]}\left\{ \va_1,\, \va_2\ \right\} = \left\{ \colvecf{1}{1}{0}{1}, \, \colvecf{0}{1}{1}{1} \right\}
Consider the subspace W={(x,y,z)∈R3  ∣  x−y+3z=0}W=\Big\{(x,y,z)\in \mathbb{R}^3 \; | \; x-y+3z=0 \Big\} of R\mathbb{R}.
Find a basis for WW that contains the vector v⃗=[0  3  1]\vec{v}=[0 \; 3 \; 1].
Find orthogonal complement of WW in R3\mathbb{R}^3.
19.4F_WML_6_$\tkco{eg7}$_$\key{Final}$_Builder_$\tkcth{17.1.}\tkct{7}$_
Find a basis for the set of all vectors that are perpendicular to both {[1102], [1010]}\left\{ \colvecf{1}{1}{0}{2}, \, \colvecf{1}{0}{1}{0} \right\}
19.4F_WML_6_$\tkco{eg13}$_$\key{Final}$_Builder_$\tkcth{17.1.}\tkct{14}$_
Show that any vector v⃗\vv in the span of the set
V={v⃗1, v⃗2, v⃗3}={[102], [214], [112]} V = \left\{ \vv_1,\, \vv_2,\, \vv_3 \right\} = \left\{ \colvecth{1}{0}{2}, \, \colvecth{2}{1}{4}, \, \colvecth{1}{1}{2} \right\}
will be perpendicular to any vector w⃗\vw in the span of the orthogonal complement W=V⊥W = V^{\perp} of the set.
19.4F_WML_6_$\tkco{eg12}$_$\key{Final}$_Builder_$\tkcth{17.1.}\tkct{13}$_
Describe the orthogonal complement of the matrix whose columns are given by the vectors {v⃗1, v⃗2, v⃗3}={[102], [214], [112]}\left\{ \vv_1,\, \vv_2,\, \vv_3 \right\} = \left\{ \colvecth{1}{0}{2}, \, \colvecth{2}{1}{4}, \, \colvecth{1}{1}{2} \right\} by finding a basis for the orthogonal complement, and describing the span of the basis in geometric terms.
Example: Fourier Expansion
Example: Fourier Expansion
Write the vector (1,1,1)(1,1,1) as a linear combination of {(1,0,1),(1,1,−1),(−1,2,1)}\{(1,0,1),(1,1,-1), (-1,2,1)\}
Basis of Orthogonal Complement
Find an orthonormal basis for the orthogonal complement of the intersection of the planes 2x+y−z=02x+y-z=0 and 4x−3y=04x-3y=0 in R3\mathbb{R}^3 .
Orthogonal Complements
Suppose UU is a subspace of Rn\mathbb{R}^n . Prove that (U⊥)⊥=U(U^\perp)^\perp=U .
Orthogonal Complement
If UU is a subspace of Rn\mathbb{R}^n , show that dim(U)+dim(U⊥)=ndim(U)+dim(U^\perp)=n .
Basis of Orthogonal Complement
Find a basis for the orthogonal complement of the intersection of the planes 2x+y−z=02x+y-z=0 and 4x−3y=04x-3y=0 in R3\mathbb{R}^3 .
Projection onto a Subspace
Suppose S:={a⃗1,...,a⃗k}S:=\{\vec a_1,...,\vec a_k\} is a set of orthonormal vectors in Rn\mathbb{R}^n , U:=span(S)U:=span(S) , and suppose that the n×kn\times k matrix AA is defined by A:=[a⃗1a⃗2...a⃗k]A:=\left[\begin{array}{rrrr}\vec a_1&\vec a_2&...&\vec a_k\end{array}\right] .
Show that AATAA^T is the matrix of the linear transformation: Rn→Rnx⃗→projUx⃗\begin{array}{l}\mathbb{R}^n\rightarrow\mathbb{R}^n\\ \vec x\rightarrow proj_U\vec x\end{array} .
More Linear Transformations Questions:
Consider the linear transformation T:R2→R3T:\mathbb{R}^2 \rightarrow \mathbb{R}^3 such that T[12]=[123]T\begin{bmatrix} 1\\2 \end{bmatrix}=\begin{bmatrix} 1\\2\\3 \end{bmatrix} and T[11]=[−2−12]T\begin{bmatrix} 1\\1 \end{bmatrix}=\begin{bmatrix} -2\\-1\\2 \end{bmatrix}
Then T[1−1]T\begin{bmatrix} 1\\-1 \end{bmatrix} is equal to
True or False: Let T:R4→R2T:\mathbb{R}^4\to\mathbb{R}^2 defined by T(x1,x2,x3,x4)=(x1+2x2,x3−x4)T(x_1,x_2,x_3,x_4)=(x_1+2x_2,x_3-x_4) , then TT is a linear transformation and the matrix of TT is given by
T=[1200001−1]T=\begin{bmatrix}1&2&0&0\\0&0&1&-1\end{bmatrix}
Determine if a transformation is linear
Let W={(x,y,z)  ∣  x,y,z∈R+}W=\{(x,y,z)\;|\;x,y,z\in\mathbb{R}^+\} be the vector with defined operations:
(x1,y1,z2)+(x2,y2,z2)=(x1x2,  y1y2,  z1z2)(x_1,y_1,z_2)+(x_2,y_2,z_2)=(x_1x_2,\;y_1y_2,\;z_1z_2)
k⋅(x,y,z)=(xk,  yk,  zk)k\cdot(x,y,z)=(x^k,\;y^k,\;z^k)
Matrix of a Linear Transformation
Consider the map T:R3→R3x⃗→proj(1,1,0)x⃗+2proj(1,0,−1)x⃗−proj(0,0,1)x⃗\begin{array}{l}T:\mathbb{R}^3\rightarrow\mathbb{R}^3\\\vec x\rightarrow proj_{(1,1,0)}\vec x+2proj_{(1,0,-1)}\vec x-proj_{(0,0,1)}\vec x\end{array}
a) Show that T is a linear transformation
b) Compute the matrix of T, and its inverse matrix.
Find the matrix of a given transformation
(a) Let L:R3⟶R4L:\mathbb{R}^3\longrightarrow\mathbb{R}^4 be defined by:
L((x,y,z))=(2x−y,2y+3z,−3x+5z,x−y+z)L\big((x,y,z)\big)=(2x-y,2y+3z,-3x+5z,x-y+z)
Find the matrix AA that represents LL
Supposse T[01]=[112],  T[11]=[21−1],  B=[010],  and  C=[001]T\begin{bmatrix} 0\\1 \end{bmatrix}=\begin{bmatrix} 1\\1\\2 \end{bmatrix},\; T\begin{bmatrix} 1\\1 \end{bmatrix}=\begin{bmatrix} 2\\1\\-1 \end{bmatrix},\; B=\begin{bmatrix} 0\\1\\0 \end{bmatrix},\; and\; C=\begin{bmatrix} 0\\0\\1 \end{bmatrix}
Then:
Finding Non-linear Transformation
Which of the following transformations are linear?
a)[xy]→[x−y2x+y+60]a)\begin{bmatrix} x\\y \end{bmatrix}→\begin{bmatrix} x-y\\ 2x+y+6\\ 0 \end{bmatrix}
Let T be a linear transformation. Suppose that
T([101])=[−20−1],T([011])=[032],T([010])=[0−40]T\left(\begin{bmatrix} 1\\0\\1 \end{bmatrix}\right)=\begin{bmatrix} -2\\0\\-1 \end{bmatrix},T\left(\begin{bmatrix} 0\\1\\1 \end{bmatrix}\right)=\begin{bmatrix} 0\\3\\2 \end{bmatrix},T\left(\begin{bmatrix} 0\\1\\0 \end{bmatrix}\right)=\begin{bmatrix} 0\\-4\\0 \end{bmatrix}
a) Find T([100])a)\ Find\ T\left(\begin{bmatrix} 1\\0\\0 \end{bmatrix}\right)
Find a linear transformation T from R2 to R2, if
T([1−4])=[−36]T\left(\begin{bmatrix} 1\\-4 \end{bmatrix}\right)=\begin{bmatrix} -3\\6 \end{bmatrix}
and it transforms points on liney=0 y=0 to new line y=2xy=2x and vice versa:
Let u⃗=[−314]\vec{u} = \begin{bmatrix} -3 \\ 1 \\ 4 \end{bmatrix} and let T:R3→R3T : \mathbb{R}^3 \rightarrow \mathbb{R}^3 be the linear transformation defined by T(v⃗)=proj⁡u⃗(v⃗)T(\vec{v}) = \operatorname{proj}_{_{\vec{u}}} (\vec{v}). Find the matrix of TT.
Let T:R2→R2T:\mathbb{R}^2\to\mathbb{R}^2 be a linear transformation with T[3−2]=[62]T\begin{bmatrix}3\\-2\end{bmatrix}=\begin{bmatrix}6\\2\end{bmatrix} and T[−54]=[−40]T\begin{bmatrix}-5\\4\end{bmatrix}=\begin{bmatrix}-4\\0\end{bmatrix}
(a) Find the matrix of TT, that is, find a matrix AA such that Tv⃗=Av⃗T\vec{v}=A\vec{v} for all v⃗∈R2\vec{v}\in\mathbb{R}^2.
(b) Is TT invertible? If so, find the matrix of T−1T^{-1}. If not, prove why not.
The unit square in R2 is the set of all points (x; y) with 0 ≤ 𝑥 ≤ 2 and 0 ≤ 𝑦 ≤ 2. Suppose each point in
the unit square undergoes a linear transformation that has the given matrix representation
A=[1221].Draw a labelled sketch of the transformed points.A=\begin{bmatrix} 1&2\\2&1 \end{bmatrix}\text{.Draw a labelled sketch of the transformed points.}
Find the matrix A representing the linear transformation T: R3 to R3 such that it satisfies:
𝑻(𝑒1) = 𝑒1 + 2𝑒2 + 3𝑒3, 𝑻(𝑒1 + 𝑒2) = 𝑒1 + 2𝑒2, 𝑻(𝑒1 + 𝑒2 + 𝑒3) = −2𝑒2 + 𝑒3.
Suppose T is a linear transformation such that
T[113]=[2−1],T[202]=[32],T[424]=[−14]T\begin{bmatrix} 1\\1\\3 \end{bmatrix}=\begin{bmatrix} 2\\-1\\ \end{bmatrix},T\begin{bmatrix} 2\\0\\2 \end{bmatrix}=\begin{bmatrix} 3\\2 \end{bmatrix},T\begin{bmatrix} 4\\2\\4 \end{bmatrix}=\begin{bmatrix} -1\\4 \end{bmatrix}
Find the matrix of T.
Let u⃗=[−311]\vec{u} = \begin{bmatrix} -3 \\ 1 \\ 1 \end{bmatrix} and let T:R3→R3T : \mathbb{R}^3\rightarrow\mathbb{R}^3 be the linear transformation defined by T(v⃗)=proj⁡u⃗(v⃗)T(\vec{v})=\operatorname{proj}_{_{\vec{u}}}(\vec{v}). Find the matrix of TT.
Find the matrix for the linear transformation S :R3 to R3 given by
S([xyz])=([−y+zx+y+z−x+2y])S\left(\begin{bmatrix} x\\y\\z \end{bmatrix}\right)=\left(\begin{bmatrix} -y+z\\ x+y+z\\ -x+2y \end{bmatrix}\right)
Suppose a⃗\vec{a}is a fixed vector in ℝ3 and consider the transformation 𝑇:ℝ3 → ℝ3 defined by 𝑇((x)⃗=a⃗ ×x.⃗𝑇(\vec{\left(x\right)}=\vec{a}\ \times\vec{x.} Prove that T is a linear transformation and find the matrix of T.
If T is a linear transformation such that T[−12]=[31]T\begin{bmatrix}-1\\2\end{bmatrix}=\begin{bmatrix}3\\1\end{bmatrix} and T[13]=[−12]T\begin{bmatrix}1\\3\end{bmatrix}=\begin{bmatrix}-1\\2\end{bmatrix} , find T[4−3]T\begin{bmatrix}4\\-3\end{bmatrix}
Suppose T is a linear transformation and
T[25]=[123],T[38]=[−101]T\begin{bmatrix} 2\\5 \end{bmatrix}=\begin{bmatrix} 1\\2\\3 \end{bmatrix},T\begin{bmatrix} 3\\8 \end{bmatrix}=\begin{bmatrix} -1\\0\\1 \end{bmatrix}
Find the matrix of this linear transformation.
Consider the transformation T defined by 𝑇(𝑥1,𝑥2,𝑥3,x4)=(x1+2x2,𝑥3−𝑥4)𝑇(𝑥_1,𝑥_2,𝑥_3,x_4)=(x_1+2x_2,𝑥_3−𝑥_4). Prove T is a linear transformation and find the matrix of T.
Suppose a⃗\vec{a} is a fixed vector in ℝn and consider the transformation 𝑇:ℝn → ℝ defined by 𝑇 (x⃗)=a.⃗  x.⃗  𝑇\ \left(\vec{x}\right)=\vec{a.}\ \ \vec{x.}\ \ Prove that T is a linear transformation and find the matrix of T.
Suppose T is a linear transformation and
T[25]=[123],T[38]=[−101]T\begin{bmatrix} 2\\5 \end{bmatrix}=\begin{bmatrix} 1\\2\\3 \end{bmatrix},T\begin{bmatrix} 3\\8 \end{bmatrix}=\begin{bmatrix} -1\\0\\1 \end{bmatrix}
Find the matrix of this linear transformation.
For a linear transformation W from R2 to R2, if 𝑊
W([20])=[32],W([12])=[−14],find W([−44]).W \left(\begin{bmatrix} 2\\0 \end{bmatrix}\right)=\begin{bmatrix} 3\\2 \end{bmatrix},W\left(\begin{bmatrix} 1\\2 \end{bmatrix}\right)=\begin{bmatrix} -1\\4 \end{bmatrix},find\ W\left(\begin{bmatrix} -4\\4 \end{bmatrix}\right).
True or False: Suppose a ⃗\vec{a\ } is a fixed vector in Rn\mathbb{R}^n and T:Rn→RT:\mathbb{R}^n\to\mathbb{R} defined by T(x⃗)=a⃗⋅x⃗T(\vec{x})=\vec{a}\cdot\vec{x} then TT is a linear transformation and its matrix is [a1⋯an]\begin{bmatrix}a_1&\cdots&a_n\end{bmatrix}
Suppose T:R2→R3T:\mathbb{R}^2\to\mathbb{R}^3 is a linear transformation and
T[25]=[123],T[38]=[−101]T\begin{bmatrix}2\\5\end{bmatrix}=\begin{bmatrix}1\\2\\3\end{bmatrix},\quad T\begin{bmatrix}3\\8\end{bmatrix}=\begin{bmatrix}-1\\0\\1\end{bmatrix}
Find the matrix of this linear transformation.
Example: Finding the Linear Transformation Matrix
Find the corresponding matrix M of the linear transformation 𝑓(𝑥, 𝑦, 𝑧) = (𝑥 − 𝑦, 3𝑦, 3𝑥 + 2𝑧).
Finding Non-linear Transformation
Which of the following transformations are linear?
Is the given transformation linear?
Let VV be the vector space {(x,y)  ∣  x,y∈R+}\{(x,y)\;|\;x,y\in\mathbb{R}^+\} with the operations:
(x1,y1)+(x2,y2)=(x1x2,y1y2)(x_1,y_1)+(x_2,y_2)=(x_1x_2,y_1y_2)
k⋅(x,y)=(xk,yk)k\cdot(x,y)=(x^k,y^k)
Practice Question: Linear and Non-Linear Maps
Determine whether the following maps are linear or not.
1. L:R→RL: \mathbb{R} \rightarrow \mathbb{R} , defined by L(x)=cos⁡(x)L(x) = \cos(x) .
2. L:R→RL : \mathbb{R} \rightarrow \mathbb{R} , defined by L(x)=3x−2L(x) = 3x -2
19.4F_Mid_Builder_$\tkcth{8.4.}\tkcf{15}$_$\tkco{Mock 3}$.
The following is known about the system of linear equations given by   ⁣M ⁣‾   x⃗=y⃗\M \, \vx = \vec{y}:
There is a unique solution for any y⃗ ∈ R5\vec{y} \, \in \, \mathbb{R}^{5}.
If one interchanges the 3rd and 4th rows and then adds twice the third row to the first before finally multiplying the 2nd and 4th rows by 3,   ⁣M ⁣‾  \M will be in its RREF.
Find the matrix of a given transformation
(a) Let L:R3⟶R4L:\mathbb{R}^3\longrightarrow\mathbb{R}^4 be defined by:
L((x,y,z))=(2x−y,2y+3z,−3x+5z,x−y+z)L\big((x,y,z)\big)=(2x-y,2y+3z,-3x+5z,x-y+z)
Find the matrix AA that represents LL
19.4F_Mid_Builder_$\tkcth{8.4.}\tkcf{14}$_
Given the transformation T:R2 → R2T: \mathbb{R}^2 \, \to \, \mathbb{R}^2 where T(v⃗)=x⃗T(\vv) = \vx and T(w⃗)=y⃗T(\vw) = \vec{y}, where
v⃗=[14]\vv = \colvec{1}{4}, w⃗=[311]\vw = \colvec{3}{11}, x⃗=[02]\vx = \colvec{0}{2} and y⃗=[−30]\vec{y} = \colvec{-3}{0},
find the matrix representation   ⁣M ⁣‾  TB\M_{T_B} of the transformation TT in the basis B={[28], [−3−11]}B = \left\{ \colvec{2}{8},\, \colvec{-3}{-11} \right\} .
Let L:R2→R3L : \mathbb{R}^2 \rightarrow \mathbb{R}^{3} be the map defined by
L(x,y)=(2x,3y,x+y)L(x, y) = (2x, 3y, x + y)
Is this map a linear transformation? If so, what is the kernel and image of this function?
133 - FML 3 - 18.1W - e.g. 34
If T[10]=[4−4]\bcb{T\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 4 \\ -4 \end{bmatrix}} and T[01]=[3−1]\bcb{T\begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ -1 \end{bmatrix}}, find the matrix representation of the transformation in the space spanned by the basis vectors u⃗1=(1,1)\bcb{\vec{u}_1 = (1,1)} and u⃗2=(1,−1)\bcb{\vec{u}_2 = (1,-1)}.
  ⁣M ⁣‾  T{u⃗1,u⃗2}=\M_{T_{\{\vu_1, \vu_2\}}} =
7
1
-5
-3
19.4F_Mid_Builder_$\tkcth{8.5.}\tkco{5}$__$\tkco{Mock 3}$
Given the basis for R2\mathbb{R}^2
v⃗=[1−1]\vv = \colvec{1}{-1} and w⃗=[2−3]\vw = \colvec{2}{-3}.
If T(v⃗)=2v⃗+3w⃗T(\vv) = 2\vv + 3\vw and T(w⃗)=v⃗+3w⃗T(\vw) = \vv + 3\vw, determine the matrix representation of the transformation TT.
Let L:R3⟶RL:\mathbb{R}^3\longrightarrow\mathbb{R} be defined by:
L((x,y,z))=3x−y+2zL\big((x,y,z)\big)=3x-y+2z
(a) Show that v→=(2,−10,−8)\overrightarrow{v}=(2,-10,-8) is in ker(L)ker(L)
What is the matrix for the linear map
L(x,y,z)=(x+y+z,2x−3z)L(x, y, z) = (x + y + z, 2x - 3z)
with respect to the standard basis?
Let LL be a linear transformation defined by:
L(x→)=Ax→L(\overrightarrow{x})=A\overrightarrow{x}
where A=[2−31−14−3]A=\left[\begin{array}{rrr} 2&-3&1\\ -1&4&-3 \end{array}\right]
Properties
Practice: Similar Matrices
Consider the matrix A=[−1421]A= \left[ \begin{array}{rrr} -1 & 4\\ 2 & 1 \end{array} \right].
Concept Clarifier
If TT and SS are linear transformations induced by the matrices
[−2024],[1−201]\begin{bmatrix}-2&0\\2&4\end{bmatrix},\quad \begin{bmatrix}1&-2\\0&1\end{bmatrix}
respectively, then find (S∘T)(x⃗)(S \circ T)(\vec{x}) where x⃗=[15]\vec{x}=\begin{bmatrix}1\\5\end{bmatrix}
133 - FML 3 - 18.1W e.g. 2.3
Determine whether the following transformation is linear:{y1= 3(x1+x2)y2=−x1\boldsymbol{ \begin{cases} y_1 &=~ 3(x_1 + x_2) \\ y_2 & = -x_1 \end{cases} }
133 - FML 3 - 18.1W e.g. 2.3
Determine whether the following transformation is linear:{y1= 1y2=1y3= 1\boldsymbol{ \begin{cases} y_1 &=~ 1 \\ y_2 & = 1 \\ y_3 &=~ 1 \end{cases} }
133 - FML 3 - 18.1W e.g. 2.2
Determine whether the following transformation is linear:{y1= x1y2=x1x2\boldsymbol{ \begin{cases} y_1 &=~ x_1 \\ y_2 & = x_1x_2 \end{cases} }
133 - FML 3 - 18.1W e.g. 2.1
Determine whether the following transformation is linear:{y1= x3y2= x3y3= x3\boldsymbol{ \begin{cases} y_1 &=~ x_3 \\ y_2 & = ~x_3 \\ y_3 &=~ x_3 \end{cases} }
Linear Transformations
Practice: Identifying Non-Linear Transformations
Select all of the following transformations that are non-linear.
Linear Transformations
Practice: Linear Transformations
Suppose T:R2→R3T:\mathbb{R}^2\to\mathbb{R}^3 is a linear transformation where
T[25]=[123],T[38]=[−101]T\begin{bmatrix}2\\5\end{bmatrix}=\begin{bmatrix}1\\2\\3\end{bmatrix},\quad T\begin{bmatrix}3\\8\end{bmatrix}=\begin{bmatrix}-1\\0\\1\end{bmatrix}.
Let L:R3⟶R2L:\mathbb{R}^3\longrightarrow\mathbb{R}^2 be defined by:
L((x,y,z))=(x+2y+3z,−x+2y+5z)L\big((x,y,z)\big)=(x+2y+3z,-x+2y+5z)
(a) Prove that LL is a linear transformation
Define the vector space V={(x,y,z)  ∣  x,y,z>0}V=\{(x,y,z)\;|\;x,y,z>0\} with the operations:
(x1,y1,z1)+(x2,y2,z2)=(x1x2,y1y2,z1+z2)(x_1,y_1,z_1)+(x_2,y_2,z_2)=(x_1x_2,y_1y_2,\textcolor{red}{z_1+z_2})
k⋅(x,y,z)=(xk,yk,kz)k\cdot(x,y,z)=(x^k,y^k,\textcolor{red}{kz})
Let LL be the linear transformation such that L(v→)=Av→L(\overrightarrow{v})=A\overrightarrow{v} where:
A=[111−10201021−1]A=\left[\begin{array}{rrr} 1&1&1\\[0.5em] -1&0&2\\[0.5em] 0&1&0\\[0.5em] 2&1&-1\\ \end{array}\right]
Find the rule that defines LL including the domain and co-domain vector spaces
Introduction to Transformations

Practice: Linear Transformations
Suppose T:R2→R3T:\mathbb{R}^2\to\mathbb{R}^3 is a linear transformation and
T[25]=[123],T[38]=[−101]T\begin{bmatrix}2\\5\end{bmatrix}=\begin{bmatrix}1\\2\\3\end{bmatrix},\quad T\begin{bmatrix}3\\8\end{bmatrix}=\begin{bmatrix}-1\\0\\1\end{bmatrix}
Find the matrix of this linear transformation.
Linear Transformations

Given that S:R2→R2S:\mathbb{R}^2 \to \mathbb{R}^2and T:R2→R2T:\mathbb{R}^2 \to \mathbb{R}^2such that
S[35]=[1−1],S[3−1]=[25]S\begin{bmatrix}3\\5\end{bmatrix}=\begin{bmatrix}1\\-1\end{bmatrix}, \quad S\begin{bmatrix}3\\-1\end{bmatrix}=\begin{bmatrix}2\\5\end{bmatrix}and T[2−2]=[10],T[11]=[25]T\begin{bmatrix}2\\-2\end{bmatrix}=\begin{bmatrix}1\\0\end{bmatrix}, \quad T\begin{bmatrix}1\\1\end{bmatrix}=\begin{bmatrix}2\\5\end{bmatrix}
Find S∘T[3−1]S \circ T\begin{bmatrix}3\\-1\end{bmatrix}
Matrix Multiplication
Find a 3 × 3 matrix M such that
  ⁣M ⁣‾  [   0−2   0]=[−1   1−1] ⁣,  ⁣M ⁣‾  [0018]=[   0−2   3] ⁣,and  ⁣M ⁣‾  [1300]=[−2   2−1]\bcb{ \M\begin{bmatrix} \ \ \ 0\\-2\\\ \ \ 0 \end{bmatrix}= \begin{bmatrix} -1\\\ \ \ 1\\-1 \end{bmatrix}\!, \hspace{0.7cm} \M\begin{bmatrix} 0\\0\\\tfrac{1}{8} \end{bmatrix}= \begin{bmatrix} \ \ \ 0\\-2\\\ \ \ 3 \end{bmatrix}\!, \hspace{0.5cm} \text{\small{and}}\hspace{0.5cm} \M\begin{bmatrix} \tfrac{1}{3}\\0\\0 \end{bmatrix}= \begin{bmatrix} -2\\\ \ \ 2\\-1 \end{bmatrix} }
Matrix of a Projection
a) Find the matrix AA so that for x⃗∈R3\vec x \in \mathbb{R}^3 the transformation mapping x⃗→proj(−13,1,12)x⃗\vec x\rightarrow proj_{(\frac{-1}{3},1,\frac{1}{2})}\vec x is the same as the transformation mapping x⃗→Ax⃗\vec x\rightarrow A\vec x
b) What is rank(A)rank(A) ?
Matrix Multiplication
e.g. Find a 3 × 3 matrix M such that
 ⁣M[   0−2   0]=[−1   1−1] ⁣, ⁣M[000]=[   0−2   3] ⁣,and M[1300]=[−2   2−1]\!M\begin{bmatrix} \ \ \ 0\\-2\\\ \ \ 0 \end{bmatrix}= \begin{bmatrix} -1\\\ \ \ 1\\-1 \end{bmatrix}\!, \hspace{0.7cm} \!M\begin{bmatrix} 0\\0\\0 \end{bmatrix}= \begin{bmatrix} \ \ \ 0\\-2\\\ \ \ 3 \end{bmatrix}\!, \hspace{0.7cm} \text{\small{and}}\ M\begin{bmatrix} \frac{1}{3}\\0\\0 \end{bmatrix}= \begin{bmatrix} -2\\\ \ \ 2\\-1 \end{bmatrix}
Given the following matrix, what can you conclude?
A=[10−2151]A = \begin{bmatrix} 1 & 0 \\ -2 & 1\\ 5 & 1 \end{bmatrix}