To specify the subspace spanned by three vectors, one can first form a matrix with the three vectors as its columns. Then, perform row reduction to obtain the reduced row-echelon form of the matrix. The non-zero rows of the reduced row-echelon form correspond to a basis for the subspace spanned by the three vectors. Finally, the subspace spanned by the three vectors can be specified by writing the basis vectors as a linear combination. **

An affine subspace is a subset of a vector space that is obtained by translating a subspace by a fixed vector. It is a flat geometric object that does not necessarily pass through the origin. On the other hand, a spanned subspace is a subspace that is formed by taking linear combinations of a set of vectors. It is the smallest subspace that contains all the vectors in the set. **

The parameter equation of the plane E spanned by the lines G and H can be expressed as: E: r = a + s * (b - a) + t * (c - a) where r is a point on the plane, a is a point on the plane, and b and c are points on the lines G and H, respectively. The parameters s and t are real numbers. This equation represents all the points on the plane E that can be reached by taking linear combinations of the direction vectors of the lines G and H. **

There could be several reasons why Blender is not rendering the animation. Some common issues could be that the render settings are not properly configured, the render layers are not set up correctly, or there may be a problem with the file paths or output settings. It is also possible that the scene is too complex for the computer to render in a timely manner, causing it to appear as if the animation is not rendering. Checking these factors and troubleshooting any errors should help resolve the issue. **

There could be several reasons why the Blender animation is not rendering. Some common issues include incorrect render settings, insufficient memory or processing power, missing or corrupted files, or a problem with the scene setup. It's important to check the render settings, ensure that all necessary files are properly linked, and make sure that the scene is set up correctly for rendering. Additionally, checking the system resources and troubleshooting any potential hardware or software issues can help identify and resolve the problem. **

To find a school internship in the field of art, animation, and illustration in Cologne, you can start by reaching out to local art schools and universities in the area. Many of these institutions have internship programs and can help connect you with potential opportunities. You can also search for art studios, animation companies, and illustration agencies in Cologne and inquire about internship opportunities directly. Networking with professionals in the industry and attending art-related events and workshops in the city can also help you discover internship openings. Additionally, utilizing online job boards and internship websites specific to the art and design industry can be a useful way to find opportunities in Cologne. **

Yes, digital art is real art. Just like traditional art forms, digital art requires skill, creativity, and vision to create. Artists use digital tools and software to express their ideas and emotions, and the resulting artwork can be just as impactful and meaningful as any other form of art. The digital medium allows for new possibilities and techniques that can produce stunning and thought-provoking pieces. Ultimately, the value of art lies in its ability to evoke emotions, provoke thoughts, and communicate messages, regardless of the medium used. **

To determine the orthogonal complement of the vector space spanned by the vectors (0, 1, 2), you first need to find a basis for the subspace spanned by these vectors. In this case, the vectors (0, 1, 2) already form a basis for the subspace. Next, you can find the orthogonal complement by finding vectors that are orthogonal to all vectors in the subspace. In this case, any vector of the form (a, b, c) where a, b, and c are not all zero and satisfy the equation a*0 + b*1 + c*2 = 0 will be orthogonal to the subspace spanned by (0, 1, 2). **