Question

True or False? No reasons needed.

(e) Suppose β and γ are bases of F n and F m, respectively. Every m × n matrix A is equal to [T] γ β for some linear transformation T: F n → F m.

(f) Recall that P(R) is the vector space of all polynomials with coefficients in R. If a linear transformation T: P(R) → P(R) is one-to-one, then T is also onto.

(g) The vector spaces R 5 and P4(R) are isomorphic.

(h) Let T,U: V → W be linear transformations and β, γ ordered bases of V and W respectively. If [T] γ β = [U] γ β , then T = U.

(i) If A and B are n × n matrices, then AB is invertible if and only if both A and B are invertible.

(j) An n×n matrix A is invertible if and only if its associated linear transformation LA : F n → F n is an isomorphism.

(k) There exists a linear transformation R 2 → R 4 that is onto.

Answer 1