Vector Spaces

In the realm of linear algebra, one will find the pivotal concept of vector spaces, also referred to as linear spaces. These mathematical constructs blend vectors via two specific operations: addition and scalar multiplication. Essential characteristics of vector spaces encompass closure, associativity, commutativity, distributivity, the presence of an additive identity and inverses, and the harmonious interaction of scalar multiplication with field multiplication.

Rewrite the System as a Vector Equality

Let $W$ be a subspace of the space $\mathbb{R}^{4}$ with standart inner product. Let $S=\left\{\left[\begin{array}{l}1 \\ 1 \\ 1 \\ 1\end{array}\right],\left[\begin{array}{c}-1 \\ 4 \\ 4 \\ 1\end{array}\right]\right\}$ be a basis for $W$. Use the Gram-Schmidt process to transform the basis $S$ into an orthonormal basis.

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Finding the Null Space

(1 point) Find a basis of the subspace of $R^{3}$ defined by the equation $-7 x_{1}+5 x_{2}-2 x_{3}=0$

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Finding the Rank

Find the rank of the matrix $ A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 5 & 1 \\ 3 & 1 & 7 \end{bmatrix} $

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Finding the Nullity

Find the nullity of the following matrix: \[ A = \begin{pmatrix} 1 & 3 & 2 \ 2 & 6 & 4\ 3 & 9 & 6 \end{pmatrix} \]

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Determining if the Vector is in the Span of the Set

QUESTION 12.1 Basis for the 2D Space Choose one $\cdot 4$ points Which set $S=\left\{\overrightarrow{v_{1}}, \overrightarrow{v_{2}}\right\}$ form a basis for $\mathbb{R}^{2}$ ? $\overrightarrow{v_{1}}=(2,1), \overrightarrow{v_{2}}=(0,0)$ $\overrightarrow{v_{1}}=(2,0), \overrightarrow{v_{2}}=(0,1)$ $\overrightarrow{v_{1}}=(1,2), \overrightarrow{v_{2}}=(4,8)$

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Vector Spaces

Rewrite the System as a Vector Equality

Finding the Null Space

Finding the Rank

Finding the Nullity

Determining if the Vector is in the Span of the Set

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