Which of the following statements is (are) true, for three matrices?
$begingroup$
Let $A, B, C$ be three matrices such that $AB = C$.
Which of the following statements is (are) true?
the columns in $C^T$ are linear combinations of the columns in $B^T$
the columns in $C$ are linear combinations of the columns in $A^T$
the columns in $C$ are linear combinations of the columns in $B$
the columns in $C^T$ are linear combinations of the columns in $A^T$
My answer:
I guess that the only option that is correct is 3 which means that The columns in C are linear combinations of the columns in B. Am I right or could someone please help me to decide of which of these that are correct?
linear-algebra matrices matrix-equations
edited Dec 19 '18 at 18:54
Lorenzo B.
1,8602520
asked Dec 19 '18 at 18:38
andersanders
615
$endgroup$
add a comment |
$begingroup$
Let $A, B, C$ be three matrices such that $AB = C$.
Which of the following statements is (are) true?
the columns in $C^T$ are linear combinations of the columns in $B^T$
the columns in $C$ are linear combinations of the columns in $A^T$
the columns in $C$ are linear combinations of the columns in $B$
the columns in $C^T$ are linear combinations of the columns in $A^T$
My answer:
I guess that the only option that is correct is 3 which means that The columns in C are linear combinations of the columns in B. Am I right or could someone please help me to decide of which of these that are correct?
linear-algebra matrices matrix-equations
edited Dec 19 '18 at 18:54
Lorenzo B.
1,8602520
asked Dec 19 '18 at 18:38
andersanders
615
$endgroup$
$begingroup$
In fact, the correct answer is 1
$endgroup$
– Omnomnomnom
Dec 19 '18 at 19:00
add a comment |
$begingroup$
Let $A, B, C$ be three matrices such that $AB = C$.
Which of the following statements is (are) true?
the columns in $C^T$ are linear combinations of the columns in $B^T$
the columns in $C$ are linear combinations of the columns in $A^T$
the columns in $C$ are linear combinations of the columns in $B$
the columns in $C^T$ are linear combinations of the columns in $A^T$
My answer:
I guess that the only option that is correct is 3 which means that The columns in C are linear combinations of the columns in B. Am I right or could someone please help me to decide of which of these that are correct?
linear-algebra matrices matrix-equations
edited Dec 19 '18 at 18:54
Lorenzo B.
1,8602520
asked Dec 19 '18 at 18:38
andersanders
615
$endgroup$
Let $A, B, C$ be three matrices such that $AB = C$.
Which of the following statements is (are) true?
the columns in $C^T$ are linear combinations of the columns in $B^T$
the columns in $C$ are linear combinations of the columns in $A^T$
the columns in $C$ are linear combinations of the columns in $B$
the columns in $C^T$ are linear combinations of the columns in $A^T$
My answer:
I guess that the only option that is correct is 3 which means that The columns in C are linear combinations of the columns in B. Am I right or could someone please help me to decide of which of these that are correct?
linear-algebra matrices matrix-equations
linear-algebra matrices matrix-equations
edited Dec 19 '18 at 18:54
Lorenzo B.
1,8602520
asked Dec 19 '18 at 18:38
andersanders
615
edited Dec 19 '18 at 18:54
Lorenzo B.
1,8602520
asked Dec 19 '18 at 18:38
andersanders
615
edited Dec 19 '18 at 18:54
Lorenzo B.
1,8602520
edited Dec 19 '18 at 18:54
Lorenzo B.
1,8602520
edited Dec 19 '18 at 18:54
Lorenzo B.
1,8602520
1,8602520
asked Dec 19 '18 at 18:38
andersanders
615
asked Dec 19 '18 at 18:38
andersanders
615
asked Dec 19 '18 at 18:38
andersanders
615
615
$begingroup$
In fact, the correct answer is 1
$endgroup$
– Omnomnomnom
Dec 19 '18 at 19:00
add a comment |
$begingroup$
In fact, the correct answer is 1
$endgroup$
– Omnomnomnom
Dec 19 '18 at 19:00
$begingroup$
In fact, the correct answer is 1
$endgroup$
– Omnomnomnom
Dec 19 '18 at 19:00
$begingroup$
In fact, the correct answer is 1
$endgroup$
– Omnomnomnom
Dec 19 '18 at 19:00
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: The matrix $AB$ has columns that are the linear combinations of the columns of $A$, and rows that are the linear combinations of the rows of $B$. To see that this is true, note that each column of $AB$ has the form
$$
pmatrix{C_1 & C_2 & C_3} pmatrix{b_1\b_2\b_3} = b_1 C_1 + b_2 C_2 + b_3 C_3
$$
where $C_i$ is the $i$th column of $A$. Similarly, each row of $AB$ as the form
$$
pmatrix{a_1 & a_2 & a_3} pmatrix{R_1\R_2\R_3} = a_1 R_1 + a_2 R_2 + a_3 R_3
$$
where $R_i$ is the $i$th row of $B$.
Of course, the columns of $A^T$ are simply the rows of $A$.
answered Dec 19 '18 at 19:04
OmnomnomnomOmnomnomnom
128k791184
$endgroup$
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: The matrix $AB$ has columns that are the linear combinations of the columns of $A$, and rows that are the linear combinations of the rows of $B$. To see that this is true, note that each column of $AB$ has the form
$$
pmatrix{C_1 & C_2 & C_3} pmatrix{b_1\b_2\b_3} = b_1 C_1 + b_2 C_2 + b_3 C_3
$$
where $C_i$ is the $i$th column of $A$. Similarly, each row of $AB$ as the form
$$
pmatrix{a_1 & a_2 & a_3} pmatrix{R_1\R_2\R_3} = a_1 R_1 + a_2 R_2 + a_3 R_3
$$
where $R_i$ is the $i$th row of $B$.
Of course, the columns of $A^T$ are simply the rows of $A$.
answered Dec 19 '18 at 19:04
OmnomnomnomOmnomnomnom
128k791184
$endgroup$
add a comment |
$begingroup$
Hint: The matrix $AB$ has columns that are the linear combinations of the columns of $A$, and rows that are the linear combinations of the rows of $B$. To see that this is true, note that each column of $AB$ has the form
$$
pmatrix{C_1 & C_2 & C_3} pmatrix{b_1\b_2\b_3} = b_1 C_1 + b_2 C_2 + b_3 C_3
$$
where $C_i$ is the $i$th column of $A$. Similarly, each row of $AB$ as the form
$$
pmatrix{a_1 & a_2 & a_3} pmatrix{R_1\R_2\R_3} = a_1 R_1 + a_2 R_2 + a_3 R_3
$$
where $R_i$ is the $i$th row of $B$.
Of course, the columns of $A^T$ are simply the rows of $A$.
answered Dec 19 '18 at 19:04
OmnomnomnomOmnomnomnom
128k791184
$endgroup$
add a comment |
$begingroup$
Hint: The matrix $AB$ has columns that are the linear combinations of the columns of $A$, and rows that are the linear combinations of the rows of $B$. To see that this is true, note that each column of $AB$ has the form
$$
pmatrix{C_1 & C_2 & C_3} pmatrix{b_1\b_2\b_3} = b_1 C_1 + b_2 C_2 + b_3 C_3
$$
where $C_i$ is the $i$th column of $A$. Similarly, each row of $AB$ as the form
$$
pmatrix{a_1 & a_2 & a_3} pmatrix{R_1\R_2\R_3} = a_1 R_1 + a_2 R_2 + a_3 R_3
$$
where $R_i$ is the $i$th row of $B$.
Of course, the columns of $A^T$ are simply the rows of $A$.
answered Dec 19 '18 at 19:04
OmnomnomnomOmnomnomnom
128k791184
$endgroup$
Hint: The matrix $AB$ has columns that are the linear combinations of the columns of $A$, and rows that are the linear combinations of the rows of $B$. To see that this is true, note that each column of $AB$ has the form
$$
pmatrix{C_1 & C_2 & C_3} pmatrix{b_1\b_2\b_3} = b_1 C_1 + b_2 C_2 + b_3 C_3
$$
where $C_i$ is the $i$th column of $A$. Similarly, each row of $AB$ as the form
$$
pmatrix{a_1 & a_2 & a_3} pmatrix{R_1\R_2\R_3} = a_1 R_1 + a_2 R_2 + a_3 R_3
$$
where $R_i$ is the $i$th row of $B$.
Of course, the columns of $A^T$ are simply the rows of $A$.
answered Dec 19 '18 at 19:04
OmnomnomnomOmnomnomnom
128k791184
answered Dec 19 '18 at 19:04
OmnomnomnomOmnomnomnom
128k791184
answered Dec 19 '18 at 19:04
OmnomnomnomOmnomnomnom
128k791184
answered Dec 19 '18 at 19:04
OmnomnomnomOmnomnomnom
128k791184
128k791184
add a comment |
add a comment |
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$begingroup$
In fact, the correct answer is 1
$endgroup$
– Omnomnomnom
Dec 19 '18 at 19:00