031 Review Part 2, Problem 10

(a) Suppose a  $6\times 8$  matrix  $A$  has 4 pivot columns. What is  ${\text{dim Nul }}A?$  Is  ${\text{Col }}A=\mathbb {R} ^{4}?$  Why or why not?

(b) If  $A$  is a  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 7\times 5}   matrix, what is the smallest possible dimension of  ${\text{Nul }}A?$

Foundations:
1. The dimension of  ${\text{Col }}A$  is equal to the number of pivots in  $A.$
2. By the Rank Theorem, if  $A$  is a  $m\times n$  matrix, then
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\text{dim Nul }}A+{\text{dim Col }}A=n.}

Solution:

(a)

Step 1:
Since  $A$  has 4 pivot columns,
${\text{dim Col }}A=4.$

Step 2:
Since  $A$  is a  $6\times 8$  matrix,  ${\text{Col }}A$  contains vectors in  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {R} ^{6}.}
Since a vector in  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {R} ^{6}}   is not a vector in  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {R} ^{4},}   we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\text{Col }}A\neq \mathbb {R} ^{4}.}

(b)

Step 1:
By the Rank Theorem, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\text{dim Nul }}A+{\text{dim Col }}A=5.}
Thus,
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\text{dim Nul }}A=5-{\text{dim Col}}A.}

Step 2:
If we want to minimize  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\text{dim Nul }}A,}   we need to maximize  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\text{dim Col }}A.}
To do this, we need to find out the maximum number of pivots in  $A.$
Since  $A$  is a  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 7\times 5}   matrix, the maximum number of pivots in  $A$  is 5.
Hence, the smallest possible value for  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\text{dim Nul }}A}   is
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {{\text{dim Nul }}A}&=&\displaystyle {5-{\text{dim Col }}A}\\&&\\&\geq &\displaystyle {5-5}\\&&\\&\geq &\displaystyle {0.}\end{array}}}

Final Answer:
(a)     Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\text{dim Col }}A=4}   and  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\text{Col }}A\neq \mathbb {R} ^{4}}
(b)     $0$

Return to Review Problems