Determine If the Vector U Is in the Column Space

question 35

Multiple Choice

Determine if the vector u is in the column space of matrix A and whether it is in the null space of A.
-Let $\mathbf { v } _ { \mathbf { 1 } } = \left[ \begin{array} { l } - 4 \\ - 1 \\ - 2 \end{array} \right] , \mathbf { v } _ { \mathbf { 2 } } = \left[ \begin{array} { r } - 3 \\ 1 \\ - 2 \end{array} \right] , \mathbf { v } _ { \mathbf { 3 } } = \left[ \begin{array} { r } 1 \\ - 5 \\ 2 \end{array} \right]$ , and $\mathrm { H } = \operatorname { Span } \left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } \right\}$ .
Note that $\mathbf { v } _ { 3 } = 2 \mathbf { v } _ { 1 } - 3 \mathbf { v } _ { 2 }$ . Which of the following sets form a basis for the subspace $H$ , i.e., which sets form an efficient spanning set containing no unnecessary vectors?
$A : \left\{ v _ { 1 } , v _ { 2 } , v _ { 3 } \right\}$
$B : \left\{ v _ { 1 } , v _ { 2 } \right\}$
$C : \left\{ v _ { 1 } , v _ { 3 } \right\}$
D: $\left\{ v _ { 2 } , v _ { 3 } \right\}$

• A) B only
• B) A only
• C) B, C, and D
• D) \(B\) and \(C\)

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Definitions:

Cost Behavior

The way in which a cost reacts to changes in the level of activity.

Fixed Cost

Costs that do not change with the level of production or output, remaining constant even as volume changes.

Relevant Range

The range of activity within which the assumptions about fixed and variable costs are valid. Beyond this range, cost behaviors may change.

Per Unit Basis

A method of cost allocation or measurement that divides a total by the number of units to find the cost per unit.