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Section 1
Vector Spaces and Bases
Show that the following vectors are linearly independent, over $\mathbf{R}$ and over $\mathbf{C}$.(a) $(1,1,1)$ and $(0,1,-1)$(b) $(1,0)$ and $(1,1)$(c) $(-1,1,0)$ and $(0,1,2)$(d) $(2,-1)$ and $(1,0)$(c) $(\pi, 0)$ and $(0,1)$(f) $(1,2)$ and $(1,3)$(g) $(1,1,0),(1,1,1)$ and $(0,1,-1)$(h) $(0,1,1),(0,2,1)$ and $(1,5,3)$
Express the given vector $X$ as a linear combination of the given vectors $A, B$ and find the coordinates of $X$ with respect to $A, B$.(a) $X=(1,0), A=(1,1), B=(0,1)$(b) $X=(2,1), A=(1,-1), B=(1,1)$(c) $X=(1,1), A=(2,1), B=(-1,0)$(d) $X=(4,3), A=(2,1), B=(-1,0)$(You may view the above vectors as elements of $\mathbf{R}^{2}$ or $\mathbf{C}^{2}$. The coordinates will be the same.)
Find the coordinates of the vector $X$ with respect to the vectors $A, B, C$.(a) $X=(1,0,0), A=(1,1,1), B=(-1,1,0), C=(1,0,-1)$(b) $X=(1,1,1), A=(0,1,-1), B=(1,1,0), C=(1,0,2)$(c) $X=(0,0,1), A=(1,1,1), B=(-1,1,0), C=(1,0,-1)$
Let $(a, b)$ and $(c, d)$ be two vectors in $K^{2}$. If $a d-b c=0$, show that they are linearly dependent. If $a d-b c \neq 0$, show that they are linearly independent.
Prove that $1, \sqrt{2}$ are linearly independent over the rational numbers.
Prove that $1, \sqrt{3}$ are linearly independent over the rational numbers.
Let $\alpha$ be a complex number. Show that $\alpha$ is rational if and only if $1, \alpha$ are linearly dependent over the rational numbers.