# Linear Algebra

• Mar 25th 2009, 03:03 PM
treetheta
Linear Algebra
1. If A and B are m x n matricies, show that
U = [X in \$\displaystyle R^n \$ | AX = BX ] is a subspace of \$\displaystyle R^n \$

2. If [X1, X2, X3,......XK] is independant, show that [X1, X1 + X2, X1 +X2 + X3,....,X1 +X2.....XK] is also independant.

3.If [Y, X1, X2, X3,.....XK] is independant show that [Y + X1, Y + X2, Y + X3,....Y+XK] is also independant

4. Let U and W denote subspace \$\displaystyle R^n \$, and assume that U is a proper subset of W. If dimU = n-1, show that either W = U or W = \$\displaystyle R^n \$

5. Let U and W denote the subspaces of \$\displaystyle R^n \$, and assume that U is a proper subset of W. If dim W = 1, show that either U = {0} or U = W

6. Let C1, C2,....CN be fixed columns in \$\displaystyle R^n \$ and define T: \$\displaystyle R^n \$ ----> \$\displaystyle R^m \$ by \$\displaystyle T([x1 *x2 ....... xN]^T = x1C1 + x2c2+.....+XnCn\$ in \$\displaystyle R^n \$. Show that T is a linear transformation and find the matrix of T
• Mar 25th 2009, 04:21 PM
HallsofIvy
Quote:

Originally Posted by treetheta
1. If A and B are m x n matricies, show that
U = [X in \$\displaystyle R^n \$ | AX = BX ] is a subspace of \$\displaystyle R^n \$

If u and v are in U, then Au= Bu and Av= Bv. So A(u+v)= Au+ Av= Bu+ Bv= B(u+v). Similarly for A(au).

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2. If [X1, X2, X3,......XK] is independant, show that [X1, X1 + X2, X1 +X2 + X3,....,X1 +X2.....XK] is also independant.
First, the correct spelling is "independ[b]e[b]nt". Now, if [X1, X1 + X2, X1 +X2 + X3,....,X1 +X2.....XK] were not independent the there would exist numbers a1, a2, ..., aK, not all 0, such that a1X1+ a2(X1+X2)+ a3(X1+ X2+ X3)+ ...+ aK(X1+ X2+ ...+XK)= 0. rewrite that to collect "like" Xs and show that would imply that [X1, X2, ..., X] are not independent.

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3.If [Y, X1, X2, X3,.....XK] is independant show that [Y + X1, Y + X2, Y + X3,....Y+XK] is also independant
Same as 2.

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4. Let U and W denote subspace \$\displaystyle R^n \$, and assume that U is a proper subset of W. If dimU = n-1, show that either W = U or W = \$\displaystyle R^n \$
Actually, since W is NOT a "proper" subset of itself, I would say this is not true!

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5. Let U and W denote the subspaces of \$\displaystyle R^n \$, and assume that U is a proper subset of W. If dim W = 1, show that either U = {0} or U = W
Same comment as for (4).

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6. Let C1, C2,....CN be fixed columns in \$\displaystyle R^n \$ and define T: \$\displaystyle R^n \$ ----> \$\displaystyle R^m \$ by \$\displaystyle T([x1 *x2 ....... xN]^T = x1C1 + x2c2+.....+XnCn\$ in \$\displaystyle R^n \$. Show that T is a linear transformation and find the matrix of T
Show that it is a linear transformation by showing that T(u+v)= T(u)+
T(v) and T(au)= aT(u). Consider the matrix having C1, C2, ..., CN as its columns, of course.