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Let \(A\) be a square matrix and let \(\|B\|\) be any norm on matrices [one example is \(\left.\|B\|=\max _{i, j}\left|b_{i j}\right|\right]\).
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Let \(A\) be a square matrix. In the following, a sequence of matrices \(C_{j}\) converges if all of its elements converge.
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Let \(A\) be a square matrix. In the following, a sequence of matrices \(C_{j}\) converges if all of its elements converge.Let \(A\) be a square matrix. In the following, a sequence of matrices \(C_{j}\) converges if all of its elements converge. Prove that the following a ...
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asked in Mathematics 23 minutes ago
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A \(3 \times 3\) real matrix need not have any real eigenvalues.
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A \(3 \times 3\) real matrix need not have any real eigenvalues.True or False - and Why? a) A \(3 \times 3\) real matrix need not have any real eigenvalues. b) If an \(n \times n\) matrix \(A\) is invertible, then ...
by MathsGee Platinum
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asked in Mathematics 55 minutes ago
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For non-zero real numbers one uses \(\frac{1}{a}-\frac{1}{b}=\frac{b-a}{a b}\). Verify the following analog for invertible matrices \(A, B\) :
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Show that \(\operatorname{trace}(A B C)=\operatorname{trace}(C A B)=\operatorname{trace}(B C A)\).
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Show that \(\operatorname{trace}(A B C)=\operatorname{trace}(C A B)=\operatorname{trace}(B C A)\).Let \(A, B\), and \(C\) be any \(n \times n\) matrices. a) Show that \(\operatorname{trace}(A B)=\operatorname{trace}(B A)\). b) Show that \(\operator ...
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asked in Mathematics 1 hour ago
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Let \(L: V \rightarrow V\) be a linear map on a vector space \(V\).
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Let \(L: V \rightarrow V\) be a linear map on a vector space \(V\).Let \(L: V \rightarrow V\) be a linear map on a vector space \(V\). a) Show that \(\operatorname{ker} L \subset \operatorname{ker} L^{2}\) and, more g ...
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asked in Mathematics 1 hour ago
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Every real upper triangular \(n \times n\) matrix \(\left(a_{i j}\right)\) with \(a_{i i}=1, i=1,2, \ldots, n\) is invertible. Proof or counterexample.

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Let \(A\) be a real matrix, not necessarily square.
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Let \(A\) be a real matrix, not necessarily square.Let \(A\) be a real matrix, not necessarily square.   a) If \(A\) is onto, show that \(A^{}\) is one-to-one. b) If \(A\) is one-to-one, show ...
by MathsGee Platinum
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asked in Mathematics 1 hour ago
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Find the dimension of \(A\) considered as a real vector space.
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Find the dimension of \(A\) considered as a real vector space.Consider the two linear transformations on the vector space \(V=\mathbf{R}^{n}\) : \(R=\) right shift: \(\left(x_{1}, \ldots, x_{n}\right) \rightarrow ...
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asked in Mathematics 1 hour ago
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Show that \(\mathcal{L}\) and \(\mathcal{R}\) are linear spaces and compute their dimensions.
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Show that \(\mathcal{L}\) and \(\mathcal{R}\) are linear spaces and compute their dimensions.Say \(A \in M(n, \mathbb{F})\) has rank \(k\). Define \ \mathcal{L}:=\{B \in M(n, \mathbb{F}) \mid B A=0\} \quad \text { and } \quad \mathcal{R}:=\{C ...
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asked in Mathematics 1 hour ago
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Give an example of a square real matrix that has rank 2 and all of whose eigenvalues are zero.
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Let \(V, W\) be two-dimensional real vector spaces, and let \(f_{1}, \ldots, f_{5}\) be linear transformations from \(V\) to \(W\).
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Let \(V, W\) be two-dimensional real vector spaces, and let \(f_{1}, \ldots, f_{5}\) be linear transformations from \(V\) to \(W\).Let \(V, W\) be two-dimensional real vector spaces, and let \(f_{1}, \ldots, f_{5}\) be linear transformations from \(V\) to \(W\). Show that there ex ...
by MathsGee Platinum
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asked in Mathematics 1 hour ago
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Let \(A\) be a square real (or complex) matrix. Then \(A\) is invertible if and only if zero is not an eigenvalue. Proof or counterexample.
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Let \(A\) be a real square matrix satisfying \(A^{17}=0\).
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Let \(A\) be a real square matrix satisfying \(A^{17}=0\).Let \(A\) be a real square matrix satisfying \(A^{17}=0\). a) Show that the matrix \(I-A\) is invertible. b) If \(B\) is an invertible matrix, is \(B ...
by MathsGee Platinum
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asked in Mathematics 2 hours ago
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Find a real \(2 \times 2\) matrix \(A\) (other than \(A=I\) ) such that \(A^{5}=I\).
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Let \(U\) and \(V\) both be two-dimensional subspaces of \(\mathbb{R}^{5}\), and let \(W=U \cap V\). Find all possible values for the dimension of \(W\).
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Let \(C(\mathbb{R})\) be the linear space of all continuous functions from \(\mathbb{R}\) to \(\mathbb{R}\).
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Let \(C(\mathbb{R})\) be the linear space of all continuous functions from \(\mathbb{R}\) to \(\mathbb{R}\).Let \(C(\mathbb{R})\) be the linear space of all continuous functions from \(\mathbb{R}\) to \(\mathbb{R}\). a) Let \(S_{c}\) be the set of different ...
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asked in Mathematics 14 hours ago
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For which real numbers \(x\) do the vectors: \((x, 1,1,1),(1, x, 1,1),(1,1, x, 1),(1,1,1, x)\) not form a basis of \(\mathbb{R}^{4}\) ?
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Which of the following sets are linear spaces?
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Which of the following sets are linear spaces?Which of the following sets are linear spaces? a) \(\left\{X=\left(x_{1}, x_{2}, x_{3}\right)\right.\) in \(\mathbb{R}^{3}\) with the property \(\lef ...
by MathsGee Platinum
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asked in Mathematics 14 hours ago
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