No, Grief said a vector in the subspace span(u,v) is of the form au+bv for scalars a and b. To say that a vector is orthogonal to a subspace means that the vector is orthogonal to each vector in that subspace.war485 said:you said that span(u,v) is in this form au+bv
You need to show that given a vector x in span(u,v), we have w.x=0 . From above, x is of the form au+bv, so you want to show that w.(au+bv)=0. This means beginning with w.(au+bv) and showing it equals 0. As Grief already said, to do so just requires distributivity and recognising that w being orthogonal to u and to v means that w.u=0 and w.v=0.war485 said:so
w . (au+bv) = 0
w . au + w . bv = 0
where a and b are any scalar numbers
and that's all? There's no more to it?
An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors. This means that the dot product of any two columns (or rows) is equal to zero, and the length of each column (or row) is equal to 1. In other words, an orthogonal matrix is a special type of matrix that preserves the magnitude and angle between vectors.
An orthogonal matrix is closely related to the concept of span because it represents a set of vectors that are linearly independent and span a particular subspace. In other words, the columns of an orthogonal matrix form a basis for the vector space they span, and any vector within that space can be expressed as a linear combination of these columns.
Yes, an orthogonal matrix can have negative values. The orthogonality property of an orthogonal matrix only requires that the dot product of the columns (or rows) is equal to zero, not the actual values of the elements. So, an orthogonal matrix can have any real number as its elements, including negative values.
To determine if a matrix is orthogonal, you can use the following criteria:
If all three criteria are met, then the matrix is orthogonal.
Orthogonal matrices have various applications in fields such as computer graphics, signal processing, and statistics. For example, in computer graphics, orthogonal matrices are used to rotate, scale, and translate objects in 3D space. In signal processing, orthogonal matrices are used to compress and decompress signals, such as images and audio. In statistics, orthogonal matrices are used to perform data transformations and analyze multivariate data.