2. List the series of matrices that would rotate the triangle ABC 90 degrees counter-clockwise around point A. -1 0 B B' 1. Apply each of following transformation matrices T to the triangle ABC and graph (show) the results. [101] -1 0 0] [10 0 1 1 0 -1 0 0 1 Lo o 16 0 0 Describe the transformation T.

Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).

Use the standard matrix for counterclockwise rotation in R2 to rotate the triangle with vertices (3,5), (5,3) and (3,0) counterclockwise 90 about the origin. Graph the triangles.

44. Consider the set of all matrices of the form, where and are real numbers, with the same rules for addition and multiplication as in. a. Show that is a ring that does not have a unity. b. Show that is not a commutative ring.

Consider the matrices below. X=[1201],Y=[1032],Z=[3412],W=[3241] Find scalars a,b, and c such that W=aX+bY+cZ. Show that there do not exist scalars a and b such that Z=aX+bY. Show that if aX+bY+cZ=0, then a=b=c=0.

If f and g are linear transformations are represented by the matrices (0 -4/2 1) and (3 3/-5 -4), what is the image of the point (-1,1) under the composite transformation g o f?

Why does every linear transformation T from R 2 to R 2 take squares to parallelograms? Rectangles also go to parallelograms (squashed if T is not invertible).

) The linear transformation T is given by the following formula T(x1, x2, x3, x4) = (x1 - x₂ + x3 x4,2x1 Determine for what values of b it is onto and for what values of b it is one-to-one. Explain briefly why. 5. 1. - x2 + bx4, x1 - 2x2 + 3x3 - 4x4).

For A =[1 20 1 ], show that the mapping adA : M2 (R) → M2 (R) defined by adA (B) =AB − BA is a linear transformation. Find bases for the kernel and the range of adA.

Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify each answer. T(X₁, X2, X3) = (x₁ - 6x2 + 4x3, X2 - 7x3) (a) Is the linear transformation one-to-one?? OD OA. T is one-to-one because T(x) = 0 has only the trivial solution. OB. T is one-to-one because the column vectors are not scalar multiples of each other. O C. T is not one-to-one because the columns the standard matrix A are linearly independent. OD. T is not one-to-one because the columns of the standard matrix A are linearly dependent. (b) Is the linear transformation onto? OA. T is not onto because the columns of the standard matrix A span R² OB. T is onto because the columns of the standard matrix A span R². O C. T is onto because the standard matrix A does not have a pivot position for every row. OD. T is not onto because the standard matrix A does not have a pivot position for every row.