To every linear transformation TT from R2 to R2, there is an associated 2×2 matrix. Match the following linear transformations with their associated matrix. 1) Reflection about the y-axis 2) Reflection about the x-axis 3) Reflection about the line y=x 4) Clockwise rotation by π2 radians 5) Counter-clockwise rotation by π2 radians 6) The projection onto the x-axis given by T(x,y)=(x,0) a) (100−1) b) (0−110) c) (01−10) d) (−1001) e) (0110) f) (1000) g) None of the above

Question

To every linear transformation TT from R2 to R2, there is an associated 2×2 matrix. Match the following linear transformations with their associated matrix.  1) Reflection about the y-axis  2) Reflection about the x-axis  3) Reflection about the line y=x  4) Clockwise rotation by π2 radians  5) Counter-clockwise rotation by π2 radians  6) The projection onto the x-axis given by T(x,y)=(x,0)  a) (100−1)  b) (0−110)  c) (01−10)  d) (−1001)  e) (0110)  f) (1000)  g) None of the above

trovabile4p · Accepted Answer

Step 1  1. Reflection about y - axis  (x,y) becomes (−x,y)  D=(−1001)  2. Reflection about x - axis  (x,y) becomes (x,−y)  A=(100−1)  3. Reflection about the line y=x  (x,y) becomes (y,x)  E=(0110)  Step 2  4. Clockwise rotation by π2 radian  C=(01−10)  5. Counter Clockwise rotation by π2 radian  B=(0−110)  6. The projection onto the x - axis given by T(x,y)=(x,0)  F=(1000)

star233 · Accepted Answer

To match the given linear transformations with their associated matrices, we need to consider how each transformation affects the standard basis vectors of R2: e1=[10] and e2=[01]. We can then express the transformation of any vector 𝐯=[xy] as a linear combination of the transformed basis vectors.Let&#039;s go through each transformation and find their associated matrices:1) Reflection about the y-axis:When reflecting a vector 𝐯 about the y-axis, the x-coordinate is negated while the y-coordinate remains the same. Thus, we have:T(e1)=[−10]andT(e2)=[10].The associated matrix is:A1=[−1010].2) Reflection about the x-axis:When reflecting a vector 𝐯 about the x-axis, the y-coordinate is negated while the x-coordinate remains the same. We have:T(e1)=[10]andT(e2)=[0−1].The associated matrix is:A2=[100−1].3) Reflection about the line y=x:When reflecting a vector 𝐯 about the line y=x, the coordinates are swapped. We have:T(e1)=[01]andT(e2)=[10].The associated matrix is:A3=[0110].4) Clockwise rotation by π/2 radians:When rotating a vector 𝐯 clockwise by π/2 radians, the coordinates are transformed as follows:T(e1)=[01]andT(e2)=[−10].The associated matrix is:A4=[0−110].5) Counter-clockwise rotation by π/2 radians:When rotating a vector 𝐯 counter-clockwise by π/2 radians, the coordinates are transformed as follows:T(e2)=[10].The associated matrix is:A5=[01−10].6) The projection onto the x-axis given by T(x, y) = (x, 0):When projecting a vector 𝐯 onto the x-axis, the y-coordinate is set to 0 while the x-coordinate remains the same. We have:T(e1)=[10]andT(e2)=[00].The associated matrix is:A6=[1000].Therefore, matching each linear transformation with its associated matrix:1) Reflection about the y-axis: A1.2) Reflection about the x-axis: A2.3) Reflection about the line y=x: A3.4) Clockwise rotation by π/2 radians: A4.5) Counter-clockwise rotation by π/2 radians: A5.6) The projection onto the x-axis: A6.

xleb123 · Accepted Answer

Answer:(1) (Option a)(2) (Option d)(3) (Option c)(4) (Option b)(5) (Option e)(6) (Option f)Explanation:To match the given linear transformations with their associated matrices, we can consider the action of each transformation on the standard basis vectors in ℝ2.1) Reflection about the y-axis:The reflection about the y-axis maps the vector (10) to (−10) and the vector (01) to (01). Therefore, the associated matrix is:(−1001)2) Reflection about the x-axis:The reflection about the x-axis maps the vector (10) to (10) and the vector (01) to (0−1). Therefore, the associated matrix is:(100−1)3) Reflection about the line y=x:The reflection about the line y=x maps the vector (10) to (01) and the vector (01) to (10). Therefore, the associated matrix is:(0110)4) Clockwise rotation by π2 radians:The clockwise rotation by π2 radians maps the vector (10) to (0−1) and the vector (01) to (10). Therefore, the associated matrix is:(0−110)5) Counter-clockwise rotation by π2 radians:The counter-clockwise rotation by π2 radians maps the vector (10) to (01) and the vector (01) to (−10). Therefore, the associated matrix is:(01−10)6) The projection onto the x-axis given by T(x,y)=(x,0):The projection onto the x-axis maps the vector (10) to (10) and the vector (01) to (00). Therefore, the associated matrix is:(1000)Now, let&#039;s match the linear transformations with their associated matrices:1) Reflection about the y-axis → Associated matrix: (−1001) (Option a)2) Reflection about the x-axis → Associated matrix: (100−1) (Option d)3) Reflection about the line y=x → Associated matrix: (0110) (Option c)4) Clockwise rotation by π2 radians → Associated matrix: (0−110) (Option b)5) Counter-clockwise rotation by π2 radians → Associated matrix: (01−10) (Option e)6) Projection onto the x-axis → Associated matrix: (1000) (Option f)

alenahelenash · Accepted Answer

To associate each linear transformation with its corresponding matrix, we can examine how each transformation affects the standard basis vectors in ℝ2. The standard basis vectors are 𝐞1=(1,0) and 𝐞2=(0,1).1) Reflection about the y-axis:When reflecting a vector about the y-axis, the x-coordinate remains the same, while the y-coordinate is negated. Therefore, 𝐞1 is mapped to 𝐞1 and 𝐞2 is mapped to −𝐞2. This corresponds to the matrix:𝐀=(100−1)(a)2) Reflection about the x-axis:When reflecting a vector about the x-axis, the y-coordinate remains the same, while the x-coordinate is negated. Therefore, 𝐞1 is mapped to −𝐞1 and 𝐞2 is mapped to 𝐞2. This corresponds to the matrix:𝐀=(−1001)(d)3) Reflection about the line y=x:When reflecting a vector about the line y=x, the x-coordinate and y-coordinate are swapped. Therefore, 𝐞1 is mapped to 𝐞2 and 𝐞2 is mapped to 𝐞1. This corresponds to the matrix:𝐀=(0110)(e)4) Clockwise rotation by π2 radians:A clockwise rotation of π2 radians in ℝ2 maps 𝐞1 to 𝐞2 and 𝐞2 to −𝐞1. This corresponds to the matrix:𝐀=(01−10)(c)5) Counter-clockwise rotation by π2 radians:A counter-clockwise rotation of π2 radians in ℝ2 maps 𝐞1 to −𝐞2 and 𝐞2 to 𝐞1. This corresponds to the matrix:𝐀=(0−110)(b)6) The projection onto the x-axis given by T(x,y)=(x,0):The projection onto the x-axis maps any vector to a vector with its y-coordinate set to 0. Therefore, both 𝐞1 and 𝐞2 are mapped to the vector (x,0), where x can be any value. This corresponds to the matrix:𝐀=(1000)(f)To summarize, the linear transformations are matched with their associated matrices as follows:1) Reflection about the y-axis → Matrix (a)2) Reflection about the x-axis → Matrix (d)3) Reflection about the line y=x → Matrix (e)4) Clockwise rotation by π2 radians → Matrix (c)5) Counter-clockwise rotation by π2 radians → Matrix (b)6) Projection onto the x-axis → Matrix (f)

To every linear transformation TT from R^{2} to R^{2}, there is an associated 2\times

Answered question

Answer & Explanation

New Questions in Linear algebra