Examples of row vectors in the following topics:
-
- 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.
- The elements (cells) of a matrix are identified by their "addresses. " Element 1,1 is the entry in the first row and first column; element 13,2 is in the 13th row and is the second element of that row.
- 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.
-
- 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.
-
- 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).
- Therefore all the elements in the null space are orthogonal to all the elements in the row space.
- In mathematical terminology, the null space and the row space are orthogonal complements of one another.
- Similarly, vectors in the left null space of a matrix are orthogonal to all the columns of this matrix.
-
- Two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations.
- The row space of a matrix is the set of all possible linear combinations of its row vectors.
- Row multiplication (scale): Multiply a row of a matrix by a nonzero constant.
- Row addition (pivot): Add to one row of a matrix some multiple of another row.
- Then, multiply the second row by 3 and then subtract the first row from the second:
-
- By convention, in a directed (i.e. asymmetric) matrix, the sender of a tie is the row and the target of the tie is the column.
- If I take all of the elements of a row (e.g. who Bob chose as friends: ---,1,1,0) I am examining the "row vector" for Bob.
- If I look only at who chose Bob as a friend (the first column, or ---,0,1,0), I am examining the "column vector" for Bob.
- It is sometimes useful to perform certain operations on row or column vectors.
- So a "vector" can be an entire matrix (1 x ... or ...x 1), or a part of a larger matrix.
-
- 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 which case the rows of $Q$ must be orthonormal vectors too.
- 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.
-
- Shifting rows and columns (if you want to rearrange the rows, you must rearrange the columns in the same way, or the matrix won't make sense for most operations) is called "permutation" of the matrix.
- Matrix permutation (Data>Permute) simply means to change the order of the rows and columns.
- These files are simply vectors (either one row, or one column) that identify which actors are to fall into which partition.
- If I wanted to group rows 1, 2, and 5 to be new rows 1, 2, and 3; and rows 3 and 4 to be new rows 4 and 5, I would enter 1 2 4 5 3.
- If you want a more complicated sort (say "all the 3's first, then all the 1's, then all the 2's) you can use an external UCINET data file to specify this as a vector (i.e. the data set would just be: 3 1 2).
-
- Now, some people reserve the term row-reduced (or row-reduced echelon) form for the matrix that also has zeros above the ones.
- Answer: Any vector of the form $(z-1,1-2z,z)$ will do.
- The only element in the null space is the zero vector.
- The row-reduced form of the matrix is
- Answer: The vectors are obviously dependent since you cannot have four linearly independent vectors in a three dimensional space.
-
- It is easy to see that the action of this matrix on a vector is to project that vector onto the one-dimensional subspace spanned by $\mathbf{v}_i$ :
- This will project any vector in $\mathbf{R}^{m}$ onto the plane spanned by $\mathbf{v}_i$ and $\mathbf{v}_j$ .
- is a projection operator onto the row space.
- This says that any vector in can be written in terms of its component in the null space and its component in the row space of .
- $\mathbf{x} = I \mathbf{x} = \left(V_r V_r ^T + V_0 V_0 ^T \right) \mathbf{x} = (\mathbf{x})_{\rm row} + (\mathbf{x})_{\rm null}
-
- 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.
- Addition of two matrices $A$$B$ and $B$ only makes sense if they have the same number of rows and columns.
- So both matrices and vectors can be thought of as vectors in the abstract sense.
- By default, a vector $\mathbf{x}$ is regarded as a column vector.
