Am I supposed to assume the dot product here?
$begingroup$
Let $W$ be a two-dimensional subspace of $mathbb{R}^3$, and consider the orthogonal projection $pi$ of $mathbb{R}^3$ onto $W$. Let $(a_i,b_i)^t$ be the coordinate vector of $pi(e_i)$, with respect to a chosen orthonormal basis of $W$. Prove that $(a_1,a_2,a_3)$ and $(b_1,b_2,b_3)$ are orthogonal unit vectors.
This specific question has already been answered on here, but it wasn't clear to me that I'm supposed to assume the form is the dot product and that $e_i$ form an orthonormal basis for $mathbb{R}^3$. Does the statement still hold true for a general symmetric form?
vector-spaces
asked Nov 24 '18 at 3:14
Miles JohnsonMiles Johnson
1928
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add a comment |
$begingroup$
Let $W$ be a two-dimensional subspace of $mathbb{R}^3$, and consider the orthogonal projection $pi$ of $mathbb{R}^3$ onto $W$. Let $(a_i,b_i)^t$ be the coordinate vector of $pi(e_i)$, with respect to a chosen orthonormal basis of $W$. Prove that $(a_1,a_2,a_3)$ and $(b_1,b_2,b_3)$ are orthogonal unit vectors.
This specific question has already been answered on here, but it wasn't clear to me that I'm supposed to assume the form is the dot product and that $e_i$ form an orthonormal basis for $mathbb{R}^3$. Does the statement still hold true for a general symmetric form?
vector-spaces
asked Nov 24 '18 at 3:14
Miles JohnsonMiles Johnson
1928
$endgroup$
$begingroup$
Unless a specific other inner product is specified, when one talks about $Bbb R^n$, they mean with the canonical dot product. And unless a specific other definition is given for $e_i$, they mean the canonical basis. But the result does hold true for other inner products and other bases orthronormal with respect to the inner product.
$endgroup$
– Paul Sinclair
Nov 24 '18 at 15:10
add a comment |
$begingroup$
Let $W$ be a two-dimensional subspace of $mathbb{R}^3$, and consider the orthogonal projection $pi$ of $mathbb{R}^3$ onto $W$. Let $(a_i,b_i)^t$ be the coordinate vector of $pi(e_i)$, with respect to a chosen orthonormal basis of $W$. Prove that $(a_1,a_2,a_3)$ and $(b_1,b_2,b_3)$ are orthogonal unit vectors.
This specific question has already been answered on here, but it wasn't clear to me that I'm supposed to assume the form is the dot product and that $e_i$ form an orthonormal basis for $mathbb{R}^3$. Does the statement still hold true for a general symmetric form?
vector-spaces
asked Nov 24 '18 at 3:14
Miles JohnsonMiles Johnson
1928
$endgroup$
Let $W$ be a two-dimensional subspace of $mathbb{R}^3$, and consider the orthogonal projection $pi$ of $mathbb{R}^3$ onto $W$. Let $(a_i,b_i)^t$ be the coordinate vector of $pi(e_i)$, with respect to a chosen orthonormal basis of $W$. Prove that $(a_1,a_2,a_3)$ and $(b_1,b_2,b_3)$ are orthogonal unit vectors.
This specific question has already been answered on here, but it wasn't clear to me that I'm supposed to assume the form is the dot product and that $e_i$ form an orthonormal basis for $mathbb{R}^3$. Does the statement still hold true for a general symmetric form?
vector-spaces
vector-spaces
asked Nov 24 '18 at 3:14
Miles JohnsonMiles Johnson
1928
asked Nov 24 '18 at 3:14
Miles JohnsonMiles Johnson
1928
asked Nov 24 '18 at 3:14
Miles JohnsonMiles Johnson
1928
asked Nov 24 '18 at 3:14
Miles JohnsonMiles Johnson
1928
asked Nov 24 '18 at 3:14
Miles JohnsonMiles Johnson
1928
1928
$begingroup$
Unless a specific other inner product is specified, when one talks about $Bbb R^n$, they mean with the canonical dot product. And unless a specific other definition is given for $e_i$, they mean the canonical basis. But the result does hold true for other inner products and other bases orthronormal with respect to the inner product.
$endgroup$
– Paul Sinclair
Nov 24 '18 at 15:10
add a comment |
$begingroup$
Unless a specific other inner product is specified, when one talks about $Bbb R^n$, they mean with the canonical dot product. And unless a specific other definition is given for $e_i$, they mean the canonical basis. But the result does hold true for other inner products and other bases orthronormal with respect to the inner product.
$endgroup$
– Paul Sinclair
Nov 24 '18 at 15:10
$begingroup$
Unless a specific other inner product is specified, when one talks about $Bbb R^n$, they mean with the canonical dot product. And unless a specific other definition is given for $e_i$, they mean the canonical basis. But the result does hold true for other inner products and other bases orthronormal with respect to the inner product.
$endgroup$
– Paul Sinclair
Nov 24 '18 at 15:10
$begingroup$
Unless a specific other inner product is specified, when one talks about $Bbb R^n$, they mean with the canonical dot product. And unless a specific other definition is given for $e_i$, they mean the canonical basis. But the result does hold true for other inner products and other bases orthronormal with respect to the inner product.
$endgroup$
– Paul Sinclair
Nov 24 '18 at 15:10
add a comment |
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$begingroup$
Unless a specific other inner product is specified, when one talks about $Bbb R^n$, they mean with the canonical dot product. And unless a specific other definition is given for $e_i$, they mean the canonical basis. But the result does hold true for other inner products and other bases orthronormal with respect to the inner product.
$endgroup$
– Paul Sinclair
Nov 24 '18 at 15:10