row space

Examples of row space in the following topics:

• A Geometrical Picture

  • 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.
  • This means that the left null space of a matrix is the orthogonal complement of the column $\mathbf{R}^{n}$ .

• Spaces Associated with a linear system Ax = y

  • Now the column space and the nullspace are generated by $A$ .
  • What about the column space and the null space of $A^T$ ?
  • These are, respectively, the row space and the left nullspace of $A$ .
  • The nullspace and row space are subspaces of $\mathbf{R}^{m}$ , while the column space and the left nullspace are subspaces of $\mathbf{R}^{n}$ .
  • We can summarize these spaces as follows:

• Eigenvectors and Orthogonal Projections

  • On the other hand, if we only include the terms in the sum associated with the $r$ nonzero singular values, then we get a projection operator onto the non-null space (which is the row space).
  • is a projection operator onto the row space.
  • is a projection operator onto the null 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}

• Matrices and Row Operations

  • Alternatively, two $m \times n$matrices are row equivalent if and only if they have the same row space.
  • The row space of a matrix is the set of all possible linear combinations of its row vectors.
  • If the rows of the matrix represent a system of linear equations, then the row space consists of all linear equations that can be deduced algebraically from those in the system.
  • Two matrices of the same size are row equivalent if and only if the corresponding homogeneous systems have the same set of solutions, or equivalently the matrices have the same null space.
  • Row addition (pivot): Add to one row of a matrix some multiple of another row.

• Examples

  • Now, some people reserve the term row-reduced (or row-reduced echelon) form for the matrix that also has zeros above the ones.
  • So the null space is is the line spanned by
  • Find the row-reduced form and the null space of the matrix
  • Find the row-reduced form and the null space of the matrix
  • The row-reduced form of the matrix is

• The Bottom of the Periodic Table

  • In Row 6, Column 3, an empty space appears between Ba and Hf.
  • The atomic number that should be here, 57, is located at the bottom of the table in the row called the Lanthanides.
  • Directly below the space in Row 6, in Row 7, is another empty space, which is filled by a row called the Actinides, also seen at the bottom of the chart.
  • By expanding the horizontal dimensions of the table, the actinide and lanthanide rows can be put into their correct relative positions.
  • The lanthanides and actinides are added as separate but connected rows, building what is called the f-block.

• Architecture in the Hellenistic Period

  • A stoa, or a covered walkway or portico, was used to bind agorae and other public spaces.
  • It was two stories tall, and had a row of rooms on the ground floor.
  • The court was also dipteral in form, edged with a double row of 108 columns 65 feet tall which surrounded the temple.
  • The structure creates a series of imposing spaces from the exterior colonnade to the oracle rooms and the interior courtyard inside of which the shrine to Apollo stood.
  • The design was eventually changed to have three rows of eight columns across the front and back of the temple and a double row of twenty on the flanks, for a total of 104 columns.

• Introduction to Elementary operations and Gaussian Elimination

  • If you have a matrix that can be derived from another matrix by a sequence of elementary operations, then the two matrices are said to be row or column equivalent.
  • because we can add 2 times row 3 of A to row 2 of A; then interchange rows 2 and 3; finally multiply row 1 by 2.
  • The first is the application of elementary operations to try to put the matrix in row-reduced form; i.e., making zero all the elements below the main diagonal (and normalizing the diagonal elements to 1).
  • Unless the matrix is very simple, calculating any of the four fundamental subspaces is probably easiest if you put the matrix in row-reduced form first.

• Simplifying Matrices With Row Operations

  • Using elementary operations, Gaussian elimination reduces matrices to row echelon form.
  • By means of a finite sequence of elementary row operations, called Gaussian elimination, any matrix can be transformed to a row echelon form.  
  • Use elementary row operations to reduce the matrix to reduced row echelon form:
  • Using elementary row operations to obtain reduced row echelon form ('rref' in the calculator) the solution to the system is revealed in the last column: $x=2, y=3, z=-1$.
  • Use elementary row operations to put a matrix in simplified form

• The "adjacency" matrix

  • This kind of a matrix is the starting point for almost all network analysis, and is called an "adjacency matrix" because it represents who is next to, or adjacent to whom in the "social space" mapped by the relations that we have measured.
  • 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.
  • This is particularly true when the rows and columns of our matrix are "super-nodes" or "blocks. " More on that in a minute.
  • 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.
  • It is sometimes useful to perform certain operations on row or column vectors.