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\title
Math 261 -- Linear Algebra\vskip 10pt
Fall 1999 (Final Exam)
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\noindent (1) What linear relation(s) must $(w_1,w_2,w_3,w_4)$ 
satisfy in order for it to be in
the range of
$T:\Bbb R^3\to \Bbb R^4$,
$T(x_1, x_2, x_3)= (3x_1 -x_2-4x_3, 2x_2 +3x_3, 3x_1-2x_2+x_3, 4x_1+x_2-x_3)?$
\vskip20pt

\noindent (2) Show that $\Bbb R^2$ has a basis $\beta$ consisting of 
eigenvectors of
the linear transformation $T:\Bbb R^2\to \Bbb R^2$, $T(x,y)=(x+2y, 
2x+y)$. Write down the matrix
$[T]_{\beta}$.

\vskip20pt

\noindent (3) Show that $\underline {similar}$ linear maps $S,T:V\to 
V$ have the same nullity.

\vskip20pt

\noindent (4) Is there an $\underline{ injective }$ linear 
transformation from the space of all
$n\times n$ symmetric real matrices into $\Bbb R^{2n}$ ? Justify your answer.

\vskip20pt

\noindent (5) Prove that two finite-dimensional vector spaces (over 
the same field) are
isomorphic if and only if they have the same dimension.


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