Challenge Problems

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{fill,stroke} }{} } } \newcommand {\tdplotshowargcolorguide }[4]{ \par \pgfmathsetmacro {\tdplotx }{#1} \pgfmathsetmacro {\tdploty }{#2} \pgfmathsetmacro {\tdplothuestep }{5} \pgfmathsetmacro {\tdplotxsize }{#3} \pgfmathsetmacro {\tdplotysize }{#4} \par \pgfmathsetmacro {\tdplotyscale }{\tdplotysize /360} \par \foreach \tdplotphi in {0,\tdplothuestep ,...,360} { \pgfmathdivide {\tdplotphi }{360} \definecolor {tdplotfillcolor}{hsb}{\pgfmathresult ,1,1} \color {tdplotfillcolor} \par \pgfmathsetmacro {\tdplotstarty }{\tdploty + \tdplotphi * \tdplotyscale } \pgfmathsetmacro {\tdplotstopy }{\tdplotstarty + \tdplothuestep * \tdplotyscale } \pgfmathsetmacro {\tdplotstartx }{\tdplotx } \pgfmathsetmacro {\tdplotstopx }{\tdplotx + \tdplotxsize } \filldraw [tdplot_screen_coords] (\tdplotstartx ,\tdplotstarty ) rectangle (\tdplotstopx ,\tdplotstopy ); } \par \pgfmathsetmacro {\tdplotstopy }{\tdploty + (360+\tdplothuestep )*\tdplotyscale } \pgfmathsetmacro {\tdplotstopx }{\tdplotx + \tdplotxsize } 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Challenge Problems for Chapter 5

Suppose $V$ has dimension $p$ and $W$ has dimension $q$ and they are each contained in a subspace, $U$ which has dimension equal to $n$ where $n>\max \left ( p,q\right ).$ What are the possibilities for the dimension of $V\cap W$ ?

Remember that a linearly independent set can be extended to form a basis.

Click the arrow to see the answer.

Let $\left \{ x_{1},\cdots ,x_{k}\right \}$ be a basis for $V\cap W.$ Then there is a basis for $V$ and $W$ which are respectively $\left \{ x_{1},\cdots ,x_{k},y_{k+1},\cdots ,y_{p}\right \} ,\ \left \{ x_{1},\cdots ,x_{k},z_{k+1},\cdots ,z_{q}\right \}$ It follows that you must have $k+p-k+q-k\leq n$ and so you must have $p+q-n\leq k$

Let $\{\vec {v}_1, \vec {v}_2, \dots , \vec {v}_n\}$ be a basis of $\RR ^n$ . Let $A$ be an $n\times n$ matrix.

(a): If $A$ is invertible, show that $\{A\vec {v}_1, A\vec {v}_2, \dots , A\vec {v}_n\}$ is a basis of $\RR ^n$ .
(b): If $\{A\vec {v}_1, A\vec {v}_2, \dots , A\vec {v}_n\}$ is a basis of $\RR ^n$ , show that $A$ is invertible.

Show that $\text {null}(A)=\text {null}(A^TA)$ for any real matrix $A$ .

Let $A$ be an $m\times n$ matrix of rank $r$ . Show that $\text {dim}(\text {null}(A))=n-r$ .

Choose a basis $\{\vec {x}_1, \vec {x}_2, \dots , \vec {x}_k\}$ of $\text {null}A$ and extend it to a basis $\{\vec {x}_1, \vec {x}_2, \dots , \vec {x}_k, \vec {z}_1, \dots , \vec {z}_m\}$ of $\RR ^n$ . Show that $\{A\vec {z}_1, \dots , A\vec {z}_m\}$ is a basis of $\text {col}(A)$ .

Let $U$ and $W$ be subspaces of $\RR ^3$ . If $\dim (U)=\dim (W)=2$ , and $U\neq W$ , show that $\dim (U\cap W)=1$ .

See Problem prb:5.27.

Bibliography

These problems came from Chapter 6 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)

W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, pp. 358–359, and p. 370.

2024-09-06 23:47:04