🔬 Tutorial problems zeta#
\(\zeta\).1#
Consider the matrix \(A\) defined by
Do the columns of this matrix form a basis of \(\mathbb{R}^3\)? Why or why not?
Check all relevant definitions and facts
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\(\zeta\).2#
Is \(\mathbb{R}^2\) a linear subspace of \(\mathbb{R}^3\)? Why or why not?
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\(\zeta\).3#
Show that if \(T \colon \mathbb{R}^K \to \mathbb{R}^N\) is a linear function then \({\bf 0} \in \mathrm{kernel}(T)\).
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\(\zeta\).4#
Let \(S\) be any nonempty subset of \(\mathbb{R}^N\) with the following two properties:
\({\bf x}, {\bf y} \in S \implies {\bf x} + {\bf y} \in S\)
\(c \in \mathbb{R}\) and \({\bf x} \in S \implies c{\bf x} \in S\)
Is \(S\) a linear subspace of \(\mathbb{R}^N\)?
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\(\zeta\).5#
If \(S\) is a linear subspace of \(\mathbb{R}^N\) then any linear combination of \(K\) elements of \(S\) is also in \(S\). Show this for the case \(K = 3\).
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\(\zeta\).6#
Let \(\{{\bf x}_1, {\bf x}_2\}\) be a linearly independent set in \(\mathbb{R}^2\) and let \(\gamma\) be a nonzero scalar. Is it true that \(\{\gamma {\bf x}_1, \gamma {\bf x}_2\}\) is also linearly independent?
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\(\zeta\).7#
Is
in the span of \(X:=\{{\bf x}_1, {\bf x}_2, {\bf x}_3\}\), where
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\(\zeta\).8#
What is the rank of the \(N \times N\) identity matrix \({\bf I}\)?
What about the upper-triangular matrix which diagonal elements are 1?
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\(\zeta\).9#
Show that if \(T: \mathbb{R}^N \to \mathbb{R}^N\) is nonsingular, i.e. linear bijection, the inverse map \(T^{-1}\) is also linear.
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