[102] 6. Find the kernel of the linear transformation whose standard matrix is B= 215. [237]

[2x1 – 3x2] () = | *1- 4x2 Let T X2 a) Show that T is a linear transformation map. b) Find a standard matrix of the map T.

Let S be a linear transformation from R³ to R2 with associated matrix Let T be a linear transformation from R2 to R2 with associated matrix Determine the matrix C of the composition To S. -1 C 2 -3 -2 4= [23]. A= -3 B= [33]

Find the B-matrix for the transformation x-Ax when B=(b₁,b₂, b3}. A= -7 -72 1 17 -4 -72-21 The B-matrix is - 18 - 5, b₁ = -4 -4 ,b₂= -3 4 b3 4 0

M = Find the matrix M of the linear transformation T: R³ → R² given by * ((:D) - T = 2x1x₂+(-7): 6x2 +7x3 (-7) * 3).

Show that the given transformation from R² to R² is linear by showing that it is a matrix transformation. F reflects a vector in the y-axis. Reflecting a vector in the y-axis means negating the x-coordinate. So -| 1819 = [x] F= = and thus F is a matrix transformation with matrix It follows that F is a linear transformation.

Write down the matrix A that represents the linear transformation f : R² → (a) R² (i.e. so that f(x) = Ax) that rotates anti-clockwise about the origin by an angle 7/2.

Let f and g be the linear transformations represented by the matrices (2-1) and (3-3), -5 -4 respectively. What is the image of the point (-1, 1) under the composite transformation go f? O (-4,1) O (-14, 25) O (-12, 6) O(-4,0) O (-15, 24)

Construct a 4×3 matrix A and vectors ~b and ~c in R4 so that ~b is in the range ofthe linear transformation with standard matrix A but ~c is not.

Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify each answer. T(X1 X2 X3) = (x₁ - 4x2 +6x3, X2-9x3) (a) Is the linear transformation one-to-one? O A. T is one-to-one because the column vectors are not scalar multiples of each other. B. T is one-to-one because T(x) = 0 has only the trivial solution. O C. T is not one-to-one because the columns of the standard matrix A are linearly independent. O D. T is not one-to-one because the columns of the standard matrix A are linearly dependent.