Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
b. |
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Definition
| b. A is row equivalent to the n x n identity matrix. |
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Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
c. |
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Definition
| c. A has n pivot positions. |
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Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
d. |
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Definition
| d. The equation Ax=0 has only the trivial solution. |
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Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
e. |
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Definition
| e. The columns of A form a linearly independent set. |
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Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
f. |
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Definition
| f. The linear transformation x |-> Ax is one-to-one. |
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Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
g. |
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Definition
| g. The equation Ax=b has at least one solution for each b in R^n. |
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Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
h. |
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Definition
| h. The columns of A span R^n. |
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Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
i. |
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Definition
| i. The linear transformation x |-> Ax maps R^n onto R^n. |
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Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
j. |
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Definition
| j. There is an n x n matrix C such that CA = I(sub n). |
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Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
k. |
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Definition
| k. There is an n x n matrix D such that AD = I(sub n). |
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Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
l. |
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Definition
| l. A^T is an invertible matrix. |
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