Short Answer

Consider the linear space $V$ of all $n \times n$ matrices for which all the vectors ${\overset{⇀}{e}}_{1}, . . . . ., {\overset{⇀}{e}}_{n}$ are eigenvectors. Describe the space $V$ (the matrices in $V$ "have a name"), and determine the dimension of $V$ .

Hence, the required dimension is n .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of the Eigenvectors

Eigenvectors are a nonzero vector that is mapped by a given linear transformation of a vector space onto a vector that is the product of a scalar multiplied by the original vector.

Step 2: Find dimension

Let’s consider the linear space $V$ or all $n \times n$ matrices which all the vectors role="math" localid="1659528394250" ${\overset{⇀}{e}}_{b}, {\overset{⇀}{e}}_{n}$ are Eigen vectors.

We want to describe the space and determine its dimension.

The set of matrices in the space $V$ spanned by the Eigen vectors ${\overset{⇀}{e}}_{b}, {\overset{⇀}{e}}_{n}$ are called diagonal matrices.

If you put all the vectors ${\overset{⇀}{e}}_{z}, {\overset{⇀}{e}}_{n}$ together in one large $n \times n$ matrix it will have only values across the diagonal, thus the name diagonal matrix.

Since all $n$ Eigen vectors span the space and are linearly independent.

Therefore, $d i m V = n$ .

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Linear Algebra With Applications

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Short Answer

Consider the linear space $V$ of all $n \times n$ matrices for which all the vectors ${\overset{⇀}{e}}_{1}, . . . . ., {\overset{⇀}{e}}_{n}$ are eigenvectors. Describe the space $V$ (the matrices in $V$ "have a name"), and determine the dimension of $V$ .

Step by Step Solution

Step 1: Definition of the Eigenvectors

Step 2: Find dimension

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Linear Algebra With Applications

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Short Answer

Consider the linear space V of all n×n matrices for which all the vectors e⇀1,.....,e⇀n are eigenvectors. Describe the space V (the matrices in V "have a name"), and determine the dimension of V.

Step by Step Solution

Step 1: Definition of the Eigenvectors

Step 2: Find dimension

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Consider the linear space $V$ of all $n \times n$ matrices for which all the vectors ${\overset{⇀}{e}}_{1}, . . . . ., {\overset{⇀}{e}}_{n}$ are eigenvectors. Describe the space $V$ (the matrices in $V$ "have a name"), and determine the dimension of $V$ .