MathsGee Homework Help Q&A - Recent questions tagged verify
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Powered by Question2AnswerLet \(V\) be the linear space of \(n \times n\) matrices with real entries. Define a linear transformation \(T: V \rightarrow V\) by the rule \(T(A)=\frac{1}{2}\left(A+A^{T}\right)\). [Here \(A^{T}\) is the matrix transpose of \(A .]\)
https://mathsgee.com/36450/linear-matrices-entries-transformation-rightarrow-transpose
Let \(V\) be the linear space of \(n \times n\) matrices with real entries. Define a linear transformation \(T: V \rightarrow V\) by the rule \(T(A)=\frac{1}{2}\left(A+A^{T}\right)\). [Here \(A^{T}\) is the matrix transpose of \(A .]\)<br />
a) Verify that \(T\) is linear.<br />
b) Describe the image of \(T\) and find it's dimension. [Try the cases \(n=2\) and \(n=3\) first.]<br />
c) Describe the image of \(T\) and find it's dimension.<br />
34<br />
d) Verify that the rank and nullity add up to what you would expect. [NOTE: This \(\operatorname{map} T\) is called the symmetrization operator.]Mathematicshttps://mathsgee.com/36450/linear-matrices-entries-transformation-rightarrow-transposeFri, 21 Jan 2022 08:18:48 +0000For non-zero real numbers one uses \(\frac{1}{a}-\frac{1}{b}=\frac{b-a}{a b}\). Verify the following analog for invertible matrices \(A, B\) :
https://mathsgee.com/36388/numbers-frac-verify-following-analog-invertible-matrices
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 \(A, B\) :<br />
\[<br />
A^{-1}-B^{-1}=A^{-1}(B-A) B^{-1}<br />
\]<br />
[The following version is also correct: \(\left.A^{-1}-B^{-1}=B^{-1}(B-A) A^{-1} .\right]\)Mathematicshttps://mathsgee.com/36388/numbers-frac-verify-following-analog-invertible-matricesFri, 21 Jan 2022 01:46:19 +0000Verify that the Cauchy-Schwarz inequality holds. \(\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)\)
https://mathsgee.com/35875/verify-that-cauchy-schwarz-inequality-holds-mathbf-mathbf
Verify that the Cauchy-Schwarz inequality holds.<br />
<br />
(b) \(\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)\)Mathematicshttps://mathsgee.com/35875/verify-that-cauchy-schwarz-inequality-holds-mathbf-mathbfFri, 14 Jan 2022 09:59:16 +0000Verify that the Cauchy–Schwarz inequality holds for $\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)$
https://mathsgee.com/35854/verify-that-cauchy-schwarz-inequality-holds-mathbf-mathbf
Verify that the Cauchy&ndash;Schwarz inequality holds for $\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)$Mathematicshttps://mathsgee.com/35854/verify-that-cauchy-schwarz-inequality-holds-mathbf-mathbfThu, 13 Jan 2022 09:13:52 +0000Verify that the Cauchy–Schwarz inequality holds for $\mathbf{u}=(4,1,1), \mathbf{v}=(1,2,3)$
https://mathsgee.com/35853/verify-that-cauchy-schwarz-inequality-holds-mathbf-mathbf
Verify that the Cauchy&ndash;Schwarz inequality holds for $\mathbf{u}=(4,1,1), \mathbf{v}=(1,2,3)$Mathematicshttps://mathsgee.com/35853/verify-that-cauchy-schwarz-inequality-holds-mathbf-mathbfThu, 13 Jan 2022 09:13:03 +0000Verify that the Cauchy–Schwarz inequality holds for $\mathbf{u}=(1,2,1,2,3), \mathbf{v}=(0,1,1,5,-2)$
https://mathsgee.com/35852/verify-that-cauchy-schwarz-inequality-holds-mathbf-mathbf
Verify that the Cauchy&ndash;Schwarz inequality holds for $\mathbf{u}=(1,2,1,2,3), \mathbf{v}=(0,1,1,5,-2)$Mathematicshttps://mathsgee.com/35852/verify-that-cauchy-schwarz-inequality-holds-mathbf-mathbfThu, 13 Jan 2022 09:12:11 +0000