Answer to Question #302622 in Linear Algebra for Utkarsh

Question

Define T : R2  R2

: T(x1

, x2

) = (– x2

, x1

).

Show that T is a linear transformation.

What is the matrix of T with respect to the

standard basis ? What is the matrix of

T with respect to the basis {v1

, v2

} of R2

,

where v1 = (1, 2), v2 = (1, – 1) ? 4

(b) Find W, where  is with respect to the

standard inner product of R4

, and

W = {(x1

, x2

, x3

, x4

)  R4|2x1 + 3x2 + 5x3 +

x4 = 0, x1 + x2 + x3 = 0}. 3

(c) Suppose U and W are subspaces of a

vector space V, where dimRV = 8. Suppose

dimRU = 4, and dimRW = 5. What are the

possible values of dimR(U  W)