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.)

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

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

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

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

]quote](vi) Yes/No...The rank of is 3[/quote]

Third time- the rows of A form a basis for the image of A which is only 3 dimensional.

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!!