# Null space, Lin. Ind. ect...

• Nov 1st 2010, 06:46 PM
tactical
Null space, Lin. Ind. ect...
The first three columns of a 3x7 matrix A are linearly independent. For each of the following statements about A, decide if that statement always holds.

(i) Yes/No...The null space of A has a non-zero element in it.
(ii) Yes/No...The rows of A are linear independent.
(iii) Yes/No...The equation Ax=b has a solution for any 3x1 column vector b.
(iv) Yes/No...Every vector in $R^7$ is a linear combination of the rows of A.
(v) Yes/No...The fourth column of A can always be written as a linear combination of the first three columns of A
(vi) Yes/No...The rank of $A^T$ is 3

I and very lost with this question. I missed a few days of class and this was the material that was covered. If there is anyone that would be able to answer these questions and give good explanations why, it would be greatly appreciated. Thanks in advance!!
• Nov 1st 2010, 07:55 PM
HallsofIvy
Quote:

Originally Posted by tactical
The first three columns of a 3x7 matrix A are linearly independent. For each of the following statements about A, decide if that statement always holds.

(i) Yes/No...The null space of A has a non-zero element in it.

A "3x7" matrix (3 rows, 7 columns) multiplies a member of a "7x1" matrix- one column with 7 entries, a member R^7. the result of such a multiplication is a "3x1" matrix- one column with 3 entries, a member of R^3. Since the domain has greater dimension than the range, there MUST be some vectors that multiply to the 0 vector. (In fact, the nullspace has dimension 7- 3= 4.)

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(ii) Yes/No...The rows of A are linear independent.
Since the first three columns are independent, the three rows must be independent.

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(iii) Yes/No...The equation Ax=b has a solution for any 3x1 column vector b.
Since the rows are independent, the range of A is all of R^3.

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(iv) Yes/No...Every vector in $R^7$ is a linear combination of the rows of A.
The rows of A form a basis for the image of A which is only 3 dimensional.

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(v) Yes/No...The fourth column of A can always be written as a linear combination of the first three columns of A
Again, the rows of A form a basis for the image of A which is only 3 dimensional.

]quote](vi) Yes/No...The rank of $A^T$ is 3[/quote]
Third time- the rows of A form a basis for the image of A which is only 3 dimensional.

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I and very lost with this question. I missed a few days of class and this was the material that was covered. If there is anyone that would be able to answer these questions and give good explanations why, it would be greatly appreciated. Thanks in advance!!