6th October 2019

stattrek

10

What is the rank of a matrix?

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The maximum number of linearly independent vectors in a matrix is equal to the number of non-zero rows in its row echelon matrix. Therefore, to find the rank of a matrix, we simply transform the matrix to its row echelon form and count the number of non-zero rows.

Similarly, you may ask, when a matrix is invertible?

If this is the case, then the matrix B is uniquely determined by A and is called the inverse of A, denoted by A⁻¹. A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is 0.

How do you know if the inverse of a matrix exists?

If the determinant of the matrix is zero, then it will not have an inverse; the matrix is then said to be singular. Only non-singular matrices have inverses. Find the inverse of the matrix A = ( 3 1 4 2 ). result should be the identity matrix I = ( 1 0 0 1 ).

Is an all zero matrix invertible?

square matrix is not invertible if at least one row or column is zero.

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What is the full rank of a matrix?

A matrix is full row rank when each of the rows of the matrix are linearly independent and full column rank when each of the columns of the matrix are linearly independent. For a square matrix these two concepts are equivalent and we say the matrix is full rank if all rows and columns are linearly independent.

2

Is dimension the same thing as rank?

The column rank of A is the dimension of the column space of A, while the row rank of A is the dimension of the row space of A. A matrix is said to have full rank if its rank equals the largest possible for a matrix of the same dimensions, which is the lesser of the number of rows and columns.

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What do you mean by the rank of a matrix?

The rank of a matrix is defined as (a) the maximum number of linearly independent column vectors in the matrix or (b) the maximum number of linearly independent row vectors in the matrix. Both definitions are equivalent. For an r x c matrix, If r is less than c, then the maximum rank of the matrix is r.

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What is the order of the matrix?

The number of rows and columns that a matrix has is called its order or its dimension. By convention, rows are listed first; and columns, second. Thus, we would say that the order (or dimension) of the matrix below is 3 x 4, meaning that it has 3 rows and 4 columns.

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What is the trace of a matrix?

The trace of an square matrix is defined to be. (1) i.e., the sum of the diagonal elements. The matrix trace is implemented in the Wolfram Language as Tr[list]. In group theory, traces are known as "group characters."

6

What is the range of the Matrix?

In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column vectors.

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What is the kernel of a matrix?

What is a "kernel" in linear algebra? A vector v is in the kernel of a matrix A if and only if Av=0. Thus, the kernel is the span of all these vectors. Similarly, a vector v is in the kernel of a linear transformation T if and only if T(v)=0. For example the kernel of this matrix (call it A)

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What is the nullity of a matrix?

The Rank Plus Nullity Theorem. Let A be a matrix. Recall that the dimension of its column space (and row space) is called the rank of A. The dimension of its nullspace is called the nullity of A. Also, the rank of this matrix, which is the number of nonzero rows in its echelon form, is 3.

9

What is the i Matrix?

Identity matrix. In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by I_n, or simply by I if the size is immaterial or can be trivially determined by the context.

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What is the null space of a matrix?

If is your matrix, the null-space is simply put, the set of all vectors such that A ⋅ v = 0 . It's good to think of the matrix as a linear transformation; if you let h ( v ) = A ⋅ v , then the null-space is again the set of all vectors that are sent to the zero vector by .

11

What is the dimension in linear algebra?

The following literature, from Friedberg's "Linear Algebra," may be of use here: Definitions. A vector space is called finite-dimensional if it has a basis consisting of a finite number of vectors. The unique number of vectors in each basis for is called the dimension of and is denoted by dim ? ( V ) .

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How can you find the dimension?

To determine the dim weight of a package, you must first measure the length, width and height of the package in inches using the longest point on each side, taking into account any bulges or misshaped sides. You then multiply those dimensions to get the cubic size of the package.

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What is the square matrix?

In mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. Square matrices are often used to represent simple linear transformations, such as shearing or rotation.

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What is the echelon form of a matrix?

Row Echelon Form. A matrix is in row echelon form (ref) when it satisfies the following conditions. The first non-zero element in each row, called the leading entry, is 1. Each leading entry is in a column to the right of the leading entry in the previous row.

15

When a matrix is not invertible?

A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is 0. Singular matrices are rare in the sense that a square matrix randomly selected from a continuous uniform distribution on its entries will almost never be singular.

16

What is a pivot column?

A pivot position in a matrix is a location. in A that corresponds to a leading 1 in the reduced echelon form of A. A pivot column is a column of A that contains a pivot position. DEFINITION: The variables corresponding to pivot columns are called basic variables.

17

What is the determinant of a matrix?

Determinant. In linear algebra, the determinant is a value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det(A), det A, or. | A.

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What is reduced echelon form of matrix?

A matrix is in row echelon form (ref) when it satisfies the following conditions. The first non-zero element in each row, called the leading entry, is 1. Each leading entry is in a column to the right of the leading entry in the previous row.

19

What does it mean to be linearly independent?

In the theory of vector spaces, a set of vectors is said to be linearly dependent if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be linearly independent.

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