5:14. Determine if the transformation T:R→ R3 defined by T(ū) = (V1, V2, V1 + vy)...
1. (a) Let T:R' R'be defined by T(x) = 5 -2. Is T a linear transformation? If so, prove that it is. If not, explain why not. (b) More generally than part (a), suppose that T:R → R is defined by T(x) = ax +b, where a and b are constants. What must be true about a and b in order for T to be a linear transformation? Explain your answer.
Determine whether the function is a linear transformation. T: R2 R3, T(x,y) = (Vx, 5xy, vy) O linear transformation not a linear transformation
1. Is T a linear transformation? Justify completely a. T:R → RP defined by T(1, y, z) = (y, 1-22, y) b. T:R + P, defined by T(a,b,c) = (a - cr? - bx +1
Example 0.1. Determine if the linear transformation T: R3 R3 defined by T(x) = 11 2 0 1 3 -1 2 x L 2011 is invertible. Additionally, is T one-to-one? Is T onto?
linear algebra Determine whether the function is a linear transformation. T: R2 R3, T(x, y) = (x,xy, vy) O linear transformation O not a linear transformation
true or false The linear transformation T:R? R? defined by T'(x,y) = (x + y,X-V) is invertible.
Problem 6: Let B = {V1, V2, ..., Un} be a set of vectors in R", and let T:R" → R" be a linear transformation such that the set {T(01), T(V2), ...,T(Un) } is basis for R". Show that B = {01, V2, ..., Un } is also a basis for R". Problem 7: Decide whether the following statement is true or false. If it is true, prove it. If it is false, give an example to show that it...
7. Let T : V → W be a linear transformation, and let v1,v2,...,vn be vectors in V. Suppose that T (v1), T (v2), . . . , T (vn) are linearly independent. Show that v1, v2, . . . , vn are linearly independent.
Let E be the plane in R3 spanned by the orthogonal vectors v1=(121)and v2=(−11−1) The reflection across E is the linear transformation R:R3→R3 defined by the formula R(x) = 2 projE(x)−x (a) Compute R(x) for x=(1260) (b) Find the eigenspace of R corresponding to the eigenvalue 1. That is, find the set of all vectors x for which R(x) =x. Justify your answer.
For real non-zero constants a and b, consider the linear transformation T: R3 + R3 defined by orthogonal reflection in the plane ay + b2 = 0 where orthogonality is defined with respect to the dot product on R3 x R3. Find in terms of a and b the numerical entries of the matrix Aſ that represents T with respect to the standard ordered basis {el, C2, C3} of R3.