Show that the given transformation from R2 to R2 is linear by showing that it is a matrix transformation. F reflects a vector in they-axis

Show that if two real symmetric matrices A, B are congruent then there is a linear change of variables to x₁,, such that qa(x₁,,xn) = qB(x₁,...,xn).

Show that T is a linear transformation by finding a matrix that implements the mapping. Note that x1, X2, ... are not vectors but are entries in vectors. T(x1 X2 X3) = (*1 -2x2 +5x3, X2 - 9x3) A= (Type an integer or decimal for each matrix element.)

Define the linear transformation T: R²-> TR² vin T (5) = (2x-²4) Find the matrix representation of the Inverse function of T.

Show that T is a linear transformation by finding a matrix that implements the mapping. Note that x1, X2, ... are not vectors but are entries in vectors. T(X1,X2.X3) = (X1 - 8x2 + 7x3, X2 - 3x3) A = (Type an integer or decimal for each matrix element.)

Show that T is a linear transformation by finding a matrix that implements the mapping. Note that X₁, X2, vectors but are entries in a vector. A = T(X₁ X₂ X3 X4) = 3x₁ + 4x₂ − 3x3 + x4 X2 (T: R→R) are not

Let A = [1 -1 -1              -1   1 -1              -1  -1  1]. Find a diagonalization if possible.

Let T : R" → R" be a linear transformation with standard matrix A. Which of the following is not a characterization of the statement “T is surjective"? OA. RREF(A) has a pivot in every row О в. Im T = R" Oc. The columns of A span R" OD. RREF(A) has a pivot in every column

8-3) show that T is a linear transformation by finding a matrix that implements the mapping. Note that x1, x2,... are not vectors but are entries in vectors a) T(x1, x2) = (2x2-3x1, x1-4x2, 0, x2) b) T(x1, x2, x3, x4) = 2x1 + 3x - 4x4 (T:R* → R)

Suppose that f : R² → R² is the linear transformation which first rotates a vector 90 degrees clock- wise and then multiplies the first coordinate by 2. Find a 2 × 2 matrix A such that f(x) = Ax for any vector x E R².