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Determine if a transformation is linear

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Wize University Linear Algebra Textbook > Linear Transformations

Linear Transformations

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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)
Note that the zero vector in WW is: 0→=(1,1,1)\overrightarrow{0}=(1,1,1)

Consider the transformations L1:W⟶WL_1:W\longrightarrow W and L2:W⟶WL_2:W\longrightarrow W defined by:
  • L1((x,y,z))=(x4,y3,z2)L_1((x,y,z))=(x^4,y^3,z^2)
  • L2((x,y,z))=(x+y,y+z,x+z)L_2((x,y,z))=(x+y,y+z,x+z)
(a) Determine if L1L_1 is a linear transformation

(b) Determine if L2L_2 is a linear transformation
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}
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}
Projection onto a subspace
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 .