91影视

Consider a p 脳 n matrix A and a q 脳 m matrix B, and form the block matrix $\mathbf{C}\mathbf{=}\left[\begin{array}{c}A\\ B\end{array}\right]$. What is the relationship between ker(A), ker(B), and ker(C)?

Let$\mathbf{貜}$is a basis of ${\mathbf{鈩�}}^{\mathbf{n}}$Is the transformation T from${\mathbf{鈩�}}^{\mathbf{n}}$to ${\mathbf{鈩�}}^{\mathbf{n}}$given by

$\mathbf{T}\mathbf{(}\mathbf{x}\mathbf{鈨�}\mathbf{)}\mathbf{=}\mathbf{[}\mathbf{x}{\mathbf{]}}_{\mathbf{貜}}$

linear? Justify your answer.

If A and B are n 脳 m matrices such that the image of A is a subset of the image of B, then there must exist an m 脳 m matrix C such that A = BC.

In Problem 46 through 55, Find all the cubics through the given points. You may use the results from Exercises 44 and 45 throughout. If there is a unique cubic, make a rough sketch of it. If there are infinitely many cubics, sketch two of them.

52.$\left(0,0\right)\mathbf{,}\left(1,0\right)\mathbf{,}\left(2,0\right)\mathbf{,}\left(0,1\right)\mathbf{,}\left(1,1\right)\mathbf{,}\left(2,1\right)\mathbf{,}\left(0,2\right)\mathbf{,}\left(1,2\right)$.

In Problem 46 through 55, Find all the cubics through the given points. You may use the results from Exercises 44 and 45 throughout. If there is a unique cubic, make a rough sketch of it. If there are infinitely many cubics, sketch two of them.

53.$\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{2}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{2}\mathbf{)}$.

Among the 3 脳 3 matrices whose entries are all 0鈥檚 and 1鈥檚, most are invertible.

Consider the basis$\mathfrak{I}\mathbf{}\mathbf{of}\mathbf{}{\mathbf{鈩�}}^{\mathbf{2}}$consisting of the vectors$\left[\begin{array}{c}1\\ 2\end{array}\right]\mathbf{}\mathbf{and}\mathbf{}\left[\begin{array}{c}3\\ 4\end{array}\right]$

.We are told that${\left[\stackrel{鈬赌}{x}\right]}_{\mathbf{I}}\mathbf{=}\left[\begin{array}{c}7\\ 11\end{array}\right]$for a certain vector$\stackrel{\mathbf{鈬赌}}{\mathbf{x}}\mathbf{}\mathbf{in}\mathbf{}{\mathbf{鈩�}}^{\mathbf{2}}$

Find

In Problem 46 through 55, Find all the cubics through the given points. You may use the results from Exercises 44 and 45 throughout. If there is a unique cubic, make a rough sketch of it. If there are infinitely many cubics, sketch two of them.

54.$\left(0,0\right)\left(1,0\right)\left(2,0\right)\left(0,1\right)\left(1,1\right)\left(2,1\right)\left(0,2\right)\left(1,2\right)\left(2,2\right)$.

Letbe the basis of${\mathrm{鈩�}}^n$consisting of the vectors$\stackrel{\mathbf{鈬赌}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{,}\stackrel{\mathbf{鈬赌}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{,}\stackrel{\mathbf{鈬赌}}{{\mathbf{v}}_{\mathbf{3}}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\stackrel{\mathbf{鈬赌}}{{\mathbf{v}}_{\mathbf{n}}}\mathbf{,}$and letlocalid="1660636061360" $\mathfrak{R}$be some other basis of ${\mathrm{鈩�}}^n$. Islocalid="1660645467599" ${\left[\stackrel{鈫�}{v_1}\right]}_{\mathbf{R}}\mathbf{,}{\left[\stackrel{鈫�}{v_2}\right]}_{\mathbf{R}}\mathbf{,}{\left[\stackrel{鈫�}{v_3}\right]}_{\mathbf{R}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\left[\stackrel{鈫�}{v_n}\right]}_{\mathbf{R}}$a basis of${\mathbf{鈩�}}^{\mathbf{n}}$ as well?

Explain.

In Problem 46 through 55, Find all the cubics through the given points. You may use the results from Exercises 44 and 45 throughout. If there is a unique cubic, make a rough sketch of it. If there are infinitely many cubics, sketch two of them.

55.$\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{2}\mathbf{)}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{)}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{2}\mathbf{)},\left(3,3\right)$.