Examples of column vectors in the following topics:
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- Rectangles have sizes that are described by the number of rows of elements and columns of elements that they contain.
- A "3 by 6" matrix has three rows and six columns; an "I by j" matrix has I rows and j columns.
- A matrix that has only one row is called a "row vector. " A matrix that has only one column is called a "column vector.
- But "rectangular" matrices are also used, as are row and column vectors.
- A three dimensional matrix has rows, columns, and "levels" or "slices. " Each "slice" has the same rows and columns as each other slice.
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- A matrix is a rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns.
- The size of a matrix is defined by the number of rows and columns that it contains.
- Matrices which have a single row are called row vectors, and those which have a single column are called column vectors.
- A matrix which has the same number of rows and columns is called a square matrix.
- For instance, $a_{2,1}$ represents the element at the second row and first column of a matrix A.
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- This means that the entries in the rows and columns for one actor are identical to those of another.
- If the matrix were symmetric, we would need only to scan pairs of rows (or columns).
- We can see the similarity of the actors if we expand the matrix in figure 13.2 by listing the row vectors followed by the column vectors for each actor as a single column, as we have in figure 13.3.
- The ties of each actor (both out and in) are now represented as a column of data.
- Concatenated row and column adjacencies for Knoke information network
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- Two matrices have the same dimensions if they have the same number of rows and columns.
- The transpose of a matrix, denoted by $A^T$ , is achieved by exchanging the columns and rows.
- Sometimes it is useful to have a special notation for the columns of a matrix.
- Namely, the inner product is a linear combination of the columns of the matrix.
- By default, a vector $\mathbf{x}$ is regarded as a column vector.
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- Any vector in the null space of a matrix, must be orthogonal to all the rows (since each component of the matrix dotted into the vector is zero).
- Similarly, vectors in the left null space of a matrix are orthogonal to all the columns of this matrix.
- This means that the left null space of a matrix is the orthogonal complement of the column $\mathbf{R}^{n}$ .
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- ., we consider what happens when there is no vector that satisfies the equations exactly.
- If $\mathbf{y}$ were in the column space of $A$ , then there would exist a vector $\mathbf{x}$ such that $A\mathbf{x}=\mathbf{y}$ .
- On the other hand, if $\mathbf{y}$ is not in the column space of $A$ a reasonable strategy is to try to find an approximate solution from within the column space.
- Now we saw in the last chapter that the outer product of a vector or matrix with itself defined a projection operator onto the subspace spanned by the vector (or columns of the matrix).
- You can either write this vector function out explicitly in terms of its components and use ordinary calculus, or you can actually differentiate the expression with respect to the vector $\mathbf{x}$ and set the result equal to zero.
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- Even if the matrix is not square, there is still a main diagonal of elements given by $A_{ii}$ where $i$ runs from 1 to the smaller of the number of rows and columns.
- In this case, each column of $Q$$\mathbf{q}_i \cdot \mathbf{q}_i = 1$ is an orthonormal vector: $\mathbf{q}_i \cdot \mathbf{q}_i = 1$ .
- Another interpretation of the matrix-vector inner product is as a mapping from one vector space to another.
- Suppose $A\in \mathbf{R}^{{n \times m}}$ , then $A$ maps vectors in $\mathbf{R}^{m}$ into vectors in $\mathbf{R}^{n}$ .
- Therefore an orthogonal matrix maps a vector into another vector of the same norm.
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- On the other hand, if the only way for this sum of vectors to be zero is for all the coefficients themselves to be zero, then we say that the vectors are linearly independent.
- Now, this linear combination of vectors can also be written as a matrix-vector inner product.
- On the other hand, we can think in terms of the compatibility of the right hand side with the columns of the matrix.
- It's intuitively clear that no two linearly independent vectors are adequate to represent an arbitrary vector in $\mathbf{R}^{3}$ .
- Conversely, since any vector in $\mathbf{R}^{3}$ can be written as a combination of the three vectors $(1,0,0)$ , $(0,1,0)$ , and $(0,0,1)$ , it is impossible to have more than three linearly independent vectors in $\mathbf{R}^{3}$ .
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- Now if we take two basis vectors $\mathbf{R}^{2}$$(1,0)$ and $(0,1)$ (any other pair of linearly independent vectors, such as $(2,0)$ and $(1,15)$ would also work) and consider all possible linear combinations of them--this is called the span of the two vectors--we will incorporate all the elements in $\mathbf{R}^{2}$ .
- A subspace of a vector space is a nonempty subset $S$ that satisfies
- $\mathbf{R}^{n}$ obviously depends on whether the columns are linearly independent or not.
- This subspace is called the column space of the matrix and is usually denoted by $R(A)$ , for "range".
- The dimension of the column space is called the rank of the matrix.
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- Each placeholder in the interval vector tells us how many of a particular interval class are in a given set class.
- That is, because there is a 1 in the second column, a pitch class set belonging to (027) will retain 1 common tone when transposed by either T2 or T10.
- Because there is a 2 in the fifth column, it will retain2 common tones when transposed by either T5 or T7.
- If an interval class vector has a tritone, it will retain twice as many common tones under tritone transposition than is indicated in the vector.
- For example, the trichord (016) has an interval vector of <100011>.
