Transcribed Image Text:**1. Which of the following are subspaces? Justify your answer in each case.**
a. \(\{ \mathbf{x} \in \mathbb{R}^2 : x_1 + x_2 = 1 \}\)
b. \(\{ \mathbf{x} \in \mathbb{R}^3 : \mathbf{x} = \begin{bmatrix} a \\ b \\ a + b \end{bmatrix} \text{ for some } a, b \in \mathbb{R} \}\)
c. \(\{ \mathbf{x} \in \mathbb{R}^3 : x_1 + 2x_2 < 0 \}\)
d. \(\{ \mathbf{x} \in \mathbb{R}^3 : x_1^2 + x_2^2 + x_3^2 = 1 \}\)
e. \(\{ \mathbf{x} \in \mathbb{R}^3 : x_1^2 + x_2^2 + x_3^2 = 0 \}\)
f. \(\{ \mathbf{x} \in \mathbb{R}^3 : x_1^2 + x_2^2 + x_3^2 = -1 \}\)
g. \(\{ \mathbf{x} \in \mathbb{R}^3 : \mathbf{x} = s \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} + t \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} \text{ for some } s, t \in \mathbb{R} \}\)
h. \(\{ \mathbf{x} \in \mathbb{R}^3 : \mathbf{x} = \begin{bmatrix} 3 \\ 0 \\ 1 \end{bmatrix} + s \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} + t \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} \text{ for some } s, t \in \mathbb{R} \}\)
i. \(\{ \mathbf{x} \in \mathbb{R}^3 : \mathbf
Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.
