Subspaces

Introduction

Subspaces: Definition and Examples

Common Types of Subspaces

Which of the following subsets of $\mathbb{R}^2$ is a subspace?

Let $V = \{(x,y) \text{ in } \mathbb{R}^2 \mid xy=0\}$, the union of the two coordinate axes. Which statement is correct?

Complete the sentence: a subset of $\mathbb{R}^n$ that contains the zero vector and is closed under addition and under scalar multiplication is called a ______ . Enter one word.

The matrix $A = \left(\begin{array}{cc} 1 & 3 \\ 2 & 6 \end{array}\right)$ has one pivot. How many vectors are in the spanning set of $\operatorname{Nul}(A)$ that the parametric vector form produces? Enter a number.

Let $A = \left(\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right)$ and $b = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$. Enter the dimension of the column space of the augmented matrix $[A\ b]$. Enter a number.

In your own words, state the single most important definition or theorem of this section, and explain what it says.