Question

Prove $\|T(x)\| = \|x\|$ for all $x \in V$ if and only if $\langle T(x),T(y)\rangle = \langle x,y\rangle$ for all $x,y \in V$.

score 223 · Answer 1 · 2020-03-11 04:12:25

223

Views

Prove $\|T(x)\| = \|x\|$ for all $x \in V$ if and only if $\langle T(x),T(y)\rangle = \langle x,y\rangle$ for all $x,y \in V$.

Published on 11 Mar 2020 - 4:12

score 69 · Answer 2 · 2020-03-11 16:33:19

69

Views

About the $y=\sum_{i=1}^k \frac{\langle y,v_i\rangle}{||v_i||^2} v_i$

Published on 11 Mar 2020 - 16:33

score 144 · Answer 3 · 2020-03-11 18:12:07

Why is the orthognal complment of $W_1 + W_2$ equal to the intersection of the orthogonal complement of $W_1$ with the orthogonal complement of $W_2$?

score 321 · Answer 4 · 2026-03-26 04:52:20

321

Views

inner product of matrix multiplication

Published on 26 Mar 2026 - 4:52

score 64 · Answer 5 · 2020-03-11 19:47:06

64

Views

Explanation of orthonormal basis and isomorphism

Published on 11 Mar 2020 - 19:47

score 33 · Answer 6 · 2020-03-12 05:37:55

33

Views

If S is orthonormal and $y \in span (S)$, then $y=\sum_{i=1}^{k}<y,v_i> v_i$.

Published on 12 Mar 2020 - 5:37

score 190 · Answer 7 · 2026-03-26 22:15:16

190

Views

Norm of inner product in dual space

Published on 26 Mar 2026 - 22:15

score 4.9k · Answer 8 · 2020-03-12 15:14:57

4.9k

Views

Definition of adjoint operator (asking for intuition)

Published on 12 Mar 2020 - 15:14

score 36 · Answer 9 · 2026-03-27 05:39:25

36

Views

Agreement of formulas for orthogonal projection onto a unit vector and non-unit vector

Published on 27 Mar 2026 - 5:39

score 70 · Answer 10 · 2026-03-26 22:19:18

70

Views

Domain of Test Function

Published on 26 Mar 2026 - 22:19

score 1k · Answer 11 · 2020-03-13 05:45:19

1k

Views

Let $V$ be an inner product space and $T\colon V\to V$ a linear operator. Prove that $R(T^{*})^{\bot}=N(T)$.

Published on 13 Mar 2020 - 5:45

score 309 · Answer 12 · 2020-03-13 16:42:58

309

Views

If $T$ is a finite-dimensional linear operator and has an eigenvector, then so does $T^{*}$.

Published on 13 Mar 2020 - 16:42

score 1.3k · Answer 13 · 2026-03-31 18:09:23

1.3k

Views

About Schur theorem

Published on 31 Mar 2026 - 18:09

score 1.2k · Answer 14 · 2020-03-14 00:01:09

1.2k

Views

About proof of theorem (T is normal if and only if there exists an orthonormal basis for V consisting of eigenvectors of T)

Published on 14 Mar 2020 - 0:01

score 1k · Answer 15 · 2020-03-14 01:49:35

1k

Views

normal operator and prove $N(T)=N(T^{*})$

Published on 14 Mar 2020 - 1:49

List Question

Prove $\|T(x)\| = \|x\|$ for all $x \in V$ if and only if $\langle T(x),T(y)\rangle = \langle x,y\rangle$ for all $x,y \in V$.

About the $y=\sum_{i=1}^k \frac{\langle y,v_i\rangle}{||v_i||^2} v_i$

Why is the orthognal complment of $W_1 + W_2$ equal to the intersection of the orthogonal complement of $W_1$ with the orthogonal complement of $W_2$?

inner product of matrix multiplication

Explanation of orthonormal basis and isomorphism

If S is orthonormal and $y \in span (S)$, then $y=\sum_{i=1}^{k}<y,v_i> v_i$.

Norm of inner product in dual space

Definition of adjoint operator (asking for intuition)

Agreement of formulas for orthogonal projection onto a unit vector and non-unit vector

Domain of Test Function

Let $V$ be an inner product space and $T\colon V\to V$ a linear operator. Prove that $R(T^{*})^{\bot}=N(T)$.

If $T$ is a finite-dimensional linear operator and has an eigenvector, then so does $T^{*}$.

About Schur theorem

About proof of theorem (T is normal if and only if there exists an orthonormal basis for V consisting of eigenvectors of T)

normal operator and prove $N(T)=N(T^{*})$

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